The paper studies the medieval tradition of the 9th century al-Khwarizmi’s handbook on algebra compared with its Latin translation by Gerard of Cremona, later translated in Italian vernacular by an anonymous Florentine abacus master, during the 14th century. This long journey along five centuries and three countries deals accurately with the mathematical contents; by means of analysis of explicit and implied elements in the three works, we also focus on the different historical backgrounds, the social condition of the authors, the (...) cultural, mindset-related and religious obstacles they had to take into consideration, while disseminating these calculation techniques, and, finally, their teaching style. (shrink)

Recent work on the imagination has stressed the epistemic significance of imaginative experiences, notably in justifying modal beliefs. An immediate problem with this is that modal beliefs appear to admit of justification through the mere exercise of rational capacities. For instance, mastery of the concepts of square, circle, and possibility should suffice to form the justified belief that a square circle is not possible, and mastery of the concepts of pig, flying, and possibility should suffice to form a justified belief (...) that a flying pig is possible. It is thus unnecessary to try to imagine a square circle or a flying pig to justify these beliefs. In this paper, I consider three ways to defend the epistemic role of imagination in modal epistemology against this challenge. One claims that modal beliefs simply admit of justification by two separate sources: rational capacities and imaginative experience. Another holds that while beliefs about logical or conceptual modality can be justified entirely by rational capacities, beliefs about metaphysical modality require imaginative experiences. The third, which I defend, is the idea that imagination is relevant in the first instance not to modal knowledge but to modal understanding: even where imaginative experience is unnecessary for the justification of modal beliefs, it is indispensable for directly grasping certain modal propositions. (shrink)

Theories of epistemically justified belief have long assumed individualism. In its extreme, or Lockean, form individualism rules out justified belief on testimony by insisting that a subject is justified in believing a proposition only if he or she possesses first-hand justification for it. The skeptical consequences of extreme individualism have led many to adopt a milder version, attributable to Hume, on which a subject is justified in believing a proposition only if he or she is justified in believing that there (...) is testimony in favor of the proposition deriving from a reliable source. I argue that this Humean individualism also leads to skepticism in a wide range of cases; it makes it impossible for a layperson to be justified on expert testimony. In addition, I argue that the apparent motivation for the Humean view, an insistence on intellectual autonomy in justification, does not succeed in motivating it. I then explore the contours of a collectivist view of justification on testimony, with special attention to the place of a subject's intellectual autonomy in such justification. I try to bring empirical results of the psychology of persuasion to bear on the epistemological issues. (shrink)

This paper responds to John Burgess's ‘Mathematics and _Bleak House_’. While Burgess's rejection of hermeneutic fictionalism is accepted, it is argued that his two main attacks on revolutionary fictionalism fail to meet their target. Firstly, ‘philosophical modesty’ should not prevent philosophers from questioning the truth of claims made within successful practices, provided that the utility of those practices as they stand can be explained. Secondly, Carnapian scepticism concerning the meaningfulness of _metaphysical_ existence claims has no force against a _naturalized_ version (...) of fictionalism, according to which our ordinary standards of scientific evidence may show that we have no reason to believe the mathematical existence claims made within the context of our mathematical and scientific theories. (shrink)

What is mathematics about? And if it is about some sort of mathematical reality, how can we have access to it? This is the problem raised by Plato, which still today is the subject of lively philosophical disputes. This book traces the history of the problem, from its origins to its contemporary treatment. It discusses the answers given by Aristotle, Proclus and Kant, through Frege's and Russell's versions of logicism, Hilbert's formalism, Gödel's platonism, up to the the current debate on (...) Benacerraf's dilemma and the indispensability argument. Through the considerations of themes in the philosophy of language, ontology, and the philosophy of science, the book aims at offering an historically-informed introduction to the philosophy of mathematics, approached through the lenses of its most fundamental problem. (shrink)

© The Authors [2018]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected] article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model...In his Moby Dick, Herman Melville writes that “to produce a mighty book you must choose a mighty theme”. Marc Lange’s Because Without Cause is definitely an impressive book that deals with a mighty theme, that of non-causal explanations in the empirical sciences and in mathematics. Blending a (...) variety of insights originating within the philosophy of science and the philosophy of mathematics, the book has an ambitious goal: capturing what features, in the many examples of putative explanations offered by mathematicians and scientists, make them genuinely explanatory.The architecture of Because Without Cause reflects its mighty character. The book is divided into four parts and eleven chapters, preceded by... (shrink)

One of the most important philosophical topics in the early twentieth century and a topic that was seminal in the emergence of analytic philosophy was the relationship between Kantian philosophy and modern geometry. This paper discusses how this question was tackled by the Neo-Kantian trained philosopher Ernst Cassirer. Surprisingly, Cassirer does not affirm the theses that contemporary philosophers often associate with Kantian philosophy of mathematics. He does not defend the necessary truth of Euclidean geometry but instead develops a kind of (...) logicism modeled on Richard Dedekind's foundations of arithmetic. Further, because he shared with other Neo-Kantians an appreciation of the developmental and historical nature of mathematics, Cassirer developed a philosophical account of the unity and methodology of mathematics over time. With its impressive attention to the detail of contemporary mathematics and its exploration of philosophical questions to which other philosophers paid scant attention, Cassirer's philosophy of mathematics surely deserves a place among the classic works of twentieth century philosophy of mathematics. Though focused on Cassirer's philosophy of geometry, this paper also addresses both Cassirer's general philosophical orientation and his reading of Kant. (shrink)

Jurado: Mario Gómez-Torrente (presidente), Miguel Ángel Fernández Vargas (vocal), Santiago Echeverri Saldarriaga (secretario). [Graduado con Mención Honorífica.].

This chapter considers the assumptions required to make scientisms of different forms genuinely threatening to philosophers, where a genuine threat would consist of a concrete risk to their statuses, the value of their teaching and research, their livelihoods, their preferred research methods, or the health of the discipline. I will find that strong and weak forms of scientism alike require substantive assumptions to make them threatening in those regards. In particular, they require sometimes heavy-handed circumscriptions of philosophy and science, as (...) well as their epistemic credentials and achievements, methods, and subject matters. They also require restrictive pronouncements upon the epistemic and non-epistemic goods that are valuable, worth promoting in academic contexts, and relevant to disciplinary health. My aim in this chapter will not to be defeat those assumptions but rather to make them explicit and to emphasize their frequent strength and contentiousness. (shrink)

Contemporary philosophy’s three main naturalisms are methodological, ontological and epistemological. Methodological naturalism states that the only authoritative standards are those of science. Ontological and epistemological naturalism respectively state that all entities and all valid methods of inquiry are in some sense natural. In philosophy of mathematics of the past few decades methodological naturalism has received the lion’s share of the attention, so we concentrate on this. Ontological and epistemological naturalism in the philosophy of mathematics are discussed more briefly in section (...) 6. (shrink)

My purpose in this paper is to examine the role of intuition in conceptual analysis and to assess whether that role can be parlayed into a plausible defense of a priori knowledge. The focus of my investigation is George Bealer’s attempt to provide such a defense. I argue that Bealer’s account of intuition and its evidential status faces three problems. I go on to examine the two primary arguments that Bealer offers against empiricism: the Starting Points Argument and the Argument (...) from Epistemic Norms. I argue that the Starting Points Argument fails because Bealer fails to show that intuitions are a priori evidence and that the Argument from Epistemic Norms fails because it is open to the Stalemate Problem. I conclude by offering an alternative approach to defending the a priori status of intuitions that avoids the Stalemate Problem. The alternative approach highlights the role of empirical investigation in defending the a priori. (shrink)

The distinction between a priori knowledge and a posteriori knowledge has come under attack in the recent literature by Philip Kitcher, John Hawthorne, C. S. Jenkins, and Timothy Williamson. Evaluating the attacks requires answering two questions. First, have they hit their target? Second, are they compelling? My goal is to argue that the attacks fail because they miss their target. Since the attacks are directed at a particular concept or distinction, they must accurately locate the target concept or distinction. Accurately (...) locating the target concept or distinction requires correctly articulating that concept or distinction. The attacks miss their target because they fail to correctly articulate the target concept or distinction. I go on to present a different challenge to the a priori-a posteriori distinction. This challenge is not directed at the coherence or significance of the distinction. Its target is the traditional view that all knowledge (or justified belief) is either a priori or a posteriori. (shrink)

The objective of this article is twofold. First, a methodological issue is addressed. It is pointed out that even if philosophers of mathematics have been recently more and more concerned with the practice of mathematics, there is still a need for a sharp definition of what the targets of a philosophy of mathematical practice should be. Three possible objects of inquiry are put forward: (1) the collective dimension of the practice of mathematics; (2) the cognitives capacities requested to the practitioners; (...) and (3) the specific forms of representation and notation shared and selected by the practitioners. Moreover, it is claimed that a broadening of the notion of ‘permissible action’ as introduced by Larvor (2012) with respect to mathematical arguments, allows for a consideration of all these three elements simultaneously. Second, a case from topology – the proof of Alexander’s theorem – is presented to illustrate a concrete analysis of a mathematical practice and to exemplify the proposed method. It is discussed that the attention to the three elements of the practice identified above brings to the emergence of philosophically relevant features in the practice of topology: the need for a revision in the definition of criteria of validity, the interest in tracking the operations that are performed on the notation, and the constant and fruitful back-and-forth from one representation to another in dealing with mathematical content. Finally, some suggestions for further research are given in the conclusions. (shrink)

This is an introduction to the volume "Explanation Beyond Causation: Philosophical Perspectives on Non-Causal Explanations", edited by A. Reutlinger and J. Saatsi (OUP, forthcoming in 2017). -/- Explanations are very important to us in many contexts: in science, mathematics, philosophy, and also in everyday and juridical contexts. But what is an explanation? In the philosophical study of explanation, there is long-standing, influential tradition that links explanation intimately to causation: we often explain by providing accurate information about the causes of the (...) phenomenon to be explained. Such causal accounts have been the received view of the nature of explanation, particularly in philosophy of science, since the 1980s. However, philosophers have recently begun to break with this causal tradition by shifting their focus to kinds of explanation that do not turn on causal information. The increasing recognition of the importance of such non-causal explanations in the sciences and elsewhere raises pressing questions for philosophers of explanation. What is the nature of non-causal explanations - and which theory best captures it? How do non-causal explanations relate to causal ones? How are non-causal explanations in the sciences related to those in mathematics and metaphysics? This volume of new essays explores answers to these and other questions at the heart of contemporary philosophy of explanation. The essays address these questions from a variety of perspectives, including general accounts of non-causal and causal explanations, as well as a wide range of detailed case studies of non-causal explanations from the sciences, mathematics and metaphysics. (shrink)

In Pantsar (2014), an outline for an empirically feasible epistemological theory of arithmetic is presented. According to that theory, arithmetical knowledge is based on biological primitives but in the resulting empirical context develops an essentially a priori character. Such contextual a priori theory of arithmetical knowledge can explain two of the three characteristics that are usually associated with mathematical knowledge: that it appears to be a priori and objective. In this paper it is argued that it can also explain the (...) ... (shrink)

Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collectionAis properly included in a collectionBthen the ‘size’ ofAshould be less than the (...) ‘size’ ofB(part–whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part–whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, envisaged by Bolzano (Kitcher). (shrink)

To explore the relation between mathematical models and reality, four different domains of reality are distinguished: observer-independent reality, personal reality, social reality and mathematical/formal reality. The concepts of personal and social reality are strongly inspired by constructivist ideas. Mathematical reality is social as well, but constructed as an autonomous system in order to make absolute agreement possible. The essential problem of mathematical modelling is that within mathematics there is agreement about ‘truth’, but the assignment of mathematics to informal reality is (...) not itself formally analysable, and it is dependent on social and personal construction processes. On these levels, absolute agreement cannot be expected. Starting from this point of view, repercussion of mathematical on social and personal reality, the historical development of mathematical modelling, and the role, use and interpretation of mathematical models in scientific practice are discussed. (shrink)

In this paper, I consider an argument for the claim that any satisfactory epistemology of mathematics will violate core tenets of naturalism, i.e. that mathematics cannot be naturalized. I find little reason for optimism that the argument can be effectively answered.

In 1902 there arrived in Jena a letter from Russell laying out a proof that shattered Frege’s confidence in logicism, which is widely taken to be the doctrine according to which every truth of arithmetic is re-expressible without relevant loss as a provable truth about a purely logical object. Frege was persuaded that Russell had exposed a pathology in logicism, which faced him with the task of examining its symptoms, diagnosing its cause, assessing its seriousness, arriving at a treatment option, (...) and making an estimate of future prospects. The symptom was the contradiction that had crept into naïve set theory in the form of the set that provably is and is not its own member. The diagnosis was that it is caused by Basic Law V of the Grundgesetze (Frege, In: Grundgesetze der Arithmetik: Begrifsschriftlich abgeleitet, Jena: Herman Pohle, 1893/1903. Translated into English as Basic Laws of Arithmetic, also edited by Philip A. Ebert and Marcus Rossberg, with Crispin Wright, Oxford: Oxford University Press, 2013). Triage answers the question, “How bad is it?” The answer was that the contradiction irreparably destroys the logicist project. The treatment option was nil. The disease was untreatable. In due course, the prognosis turned out to be that a scaled-down Fregean logic could have an honourable life as a theory inference for various domains of mathematical discourse, but not for domains containing the logical objects required for logicism. Since there aren’t such objects, there aren’t such domains. On the face of it, Frege’s logicist collapse is astonishing. Why wouldn’t he have repaired the fault in Law V and gone back to the business of bringing logicism to an assured realization? In the course of our reflections, we will have nice occasion to consider the good it might have done Frege to have booked some time with Aristotle had he been able to. By the time we’re finished, we’ll have cause to think that in the end the Russell might well have begged the question against Frege. (shrink)

In this article I discuss the 'procedural postulationist' view of mathematics advanced by Kit Fine in a recent paper. I argue that he has not shown that this view provides an avenue to knowledge of mathematical truths, at least if such truths are objective truths. In particular, more needs to be said about the criteria which constrain which types of entities can be postulated. I also argue that his reliance on second-order quantification means that his background logic is not free (...) of ontological commitment and that his doctrine of 'creative expansion' only makes sense from a radically anti-realist perspective. (shrink)

The Principal Principle (PP) says that, for any proposition A, given any admissible evidence and the proposition that the chance of A is x%, one's conditional credence in A should be x%. Humean Supervenience (HS) claims that, among possible worlds like ours, no two differ without differing in the spacetime-point-by-spacetime-point arrangement of local properties. David Lewis (1986b, 1994a) has argued that PP contradicts HS, and the validity of his argument has been endorsed by Bigelow et al. (1993), Thau (1994), Hall (...) (1994), Strevens (1995), Ismael (1996), Hoefer (1997), and Black (1998). Against this consensus, I argue that PP might not contradict HS: Lewis's argument is invalid, and every attempt – within a broad class of attempts – to amend the argument fails. (shrink)

During the first half of the twentieth century, mainstream answers to the foundational crisis, mainly triggered by Russell and Gödel, remained largely perfectibilist in nature. Along with a general naturalist wave in the philosophy of science, during the second half of that century, this idealist picture was finally challenged and traded in for more realist ones. Next to the necessary preliminaries, the present paper proposes a structured view of various philosophical accounts of mathematics indebted to this general idea, laying the (...) ground for a desirable integration of their strenghts. (shrink)

Some recent discussions of a priori knowledge, taking their departure from Kant's characterization of such knowledge as being absolutely independent of experience, have concluded that while one might delineate a concept of a priori knowledge, it fails to have any application as any purported case of such knowledge can be undermined by suitably recalcitrant experiences. In response, certain defenders of apriority have claimed that a priori justification only requires that a belief be positively dependent on no experience. In this paper, (...) I begin by showing how the exchange of arguments between the disputants comes down, in the end, to no more than a display of conflicting intuitions. I shall then provide a diagnosis explaining how our explication of a priori justification depends on our standards for applying the term 'a priori' with this, in turn, reflecting our prior intentions as to whether we are willing to allow the existence of such warrants. I shall further argue that the claim that such knowledge can be affected by subversive experience is not entirely compatible with the spirit of apriority. Finally I conclude by making some methodological remarks about the prospects of a positive characterization of a priori knowledge by comparing it to the concept of knowledge. (shrink)

According to quasi-empiricism, mathematics is very like a branch of natural science. But if mathematics is like a branch of science, and science studies real objects, then mathematics should study real objects. Thus a quasi-empirical account of mathematics must answer the old epistemological question: How is knowledge of abstract objects possible? This paper attempts to show how it is possible.The second section examines the problem as it was posed by Benacerraf in Mathematical Truth and the next section presents a way (...) of looking at abstract objects that purports to demythologize them. In particular, it shows how we can have empirical knowledge of various abstract objects and even how we might causally interact with them. (shrink)

It is argued that the philosophical and epistemological beliefs about the nature of mathematics have a significant influence on the way mathematics is taught at school. In this paper, the philosophy of mathematics of the NCTM's Standards is investigated by examining is explicit assumptions regarding the teaching and learning of school mathematics. The main conceptual tool used for this purpose is the model of two dichotomous philosophies of mathematics-absolutist versus- fallibilist and their relation to mathematics pedagogy. The main conclusion is (...) that a fallibilist view of mathematics is assumed in the Standards and that most of its pedagogical assumptions and approaches are based on this philosophy. (shrink)

Great intuitions are fundamental to conjecture and discovery in mathematics. In this paper, we investigate the role that intuition plays in mathematical thinking. We review key events in the history of mathematics where paradoxes have emerged from mathematicians' most intuitive concepts and convictions, and where the resulting difficulties led to heated controversies and debates. Examples are drawn from Riemannian geometry, set theory and the analytic theory of the continuum, and include the Continuum Hypothesis, the Tarski-Banach Paradox, and several works by (...) Gödel, Cantor, Wittgenstein and Weierstrass. We examine several fallacies of intuition and determine how far our intuitive conjectures are limited by the nature of our sense-experience, and by our capacities for conceptualization. Finally, I suggest how we can use visual and formal heuristics to cultivate our mathematical intuitions and how the breadth of this new epistemic perspective can be useful in cases where intuition has traditionally been regarded as out of its depth. (shrink)

This paper provides a computational characterization of coherence that applies to a wide range of philosophical problems and psychological phenomena. Maximizing coherence is a matter of maximizing satisfaction of a set of positive and negative constraints. After comparing five algorithms for maximizing coherence, we show how our characterization of coherence overcomes traditional philosophical objections about circularity and truth.

W. V. Quine famously claimed that no statement is immune to revision. This thesis has had a profound impact on twentieth century philosophy, and it still occupies centre stage in many contemporary debates. However, despite its importance it is not clear how it should be interpreted. I show that the thesis is in fact ambiguous between three substantially different theses. I illustrate the importance of clarifying it by assessing its use in the debate against the existence of a priori knowledge. (...) I show how the three different readings of the thesis can be used to generate three substantially different and philosophically significant arguments against the a priori. I further challenge each one of these arguments against the a priori. (shrink)

This paper develops some respects in which the philosophy of mathematics can fruitfully be informed by mathematical practice, through examining Frege's Grundlagen in its historical setting. The first sections of the paper are devoted to elaborating some aspects of nineteenth century mathematics which informed Frege's early work. (These events are of considerable philosophical significance even apart from the connection with Frege.) In the middle sections, some minor themes of Grundlagen are developed: the relationship Frege envisions between arithmetic and geometry and (...) the way in which the study of reasoning is to illuminate this. In the final section, it is argued that the sorts of issues Frege attempted to address concerning the character of mathematical reasoning are still in need of a satisfying answer. (shrink)

The distinction between a priori and a posteriori knowledge has been the subject of an enormous amount of discussion, but the literature is biased against recognizing the intimate relationship between these forms of knowledge. For instance, it seems to be almost impossible to find a sample of pure a priori or a posteriori knowledge. In this paper, it will be suggested that distinguishing between a priori and a posteriori is more problematic than is often suggested, and that a priori and (...) a posteriori resources are in fact used in parallel. We will define this relationship between a priori and a posteriori knowledge as the bootstrapping relationship. As we will see, this relationship gives us reasons to seek for an altogether novel definition of a priori and a posteriori knowledge. Specifically, we will have to analyse the relationship between a priori knowledge and a priori reasoning , and it will be suggested that the latter serves as a more promising starting point for the analysis of aprioricity. We will also analyse a number of examples from the natural sciences and consider the role of a priori reasoning in these examples. The focus of this paper is the analysis of the concepts of a priori and a posteriori knowledge rather than the epistemic domain of a posteriori and a priori justification. (shrink)

If the computational theory of mind is right, then minds are realized by machines. There is an ordered complexity hierarchy of machines. Some finite machines realize finitely complex minds; some Turing machines realize potentially infinitely complex minds. There are many logically possible machines whose powers exceed the Church–Turing limit (e.g. accelerating Turing machines). Some of these supermachines realize superminds. Superminds perform cognitive supertasks. Their thoughts are formed in infinitary languages. They perceive and manipulate the infinite detail of fractal objects. They (...) have infinitely complex bodies. Transfinite games anchor their social relations. (shrink)

For the past hundred years, mathematics, for its own reasons, has been shifting away from the study of “mathematical objects” and towards the study of “structures”. One would have expected philosophers to jump onto the bandwagon, as in many other cases, to proclaim that this shift is no accident, since mathematics is “essentially” about structures, not objects. In fact, structuralism has not been a very popular philosophy of mathematics, probably because of the hostility of Frege and other influential logicists, and (...) quasi-logicists like Quine. Recently, however, structuralism has finally penetrated the philosophical community and a number of “structuralist” volumes in the philosophy of mathematics have seen the light of day. Michael Resnik is one of the most distinguished representatives of this school, along with Stewart Shapiro, who has also recently published a structuralist tract. (shrink)

In this paper, I argue that the success of evolutionary debunking arguments hinges on what theory of epistemic justification one endorses. More specifically, I argue that what it takes to satisfactorily respond to evolutionary debunking arguments depends on what view of epistemic justification one is operating under and that a thorough analysis of any line of response to evolutionary debunking arguments must take into account whether there is a specific view of justification motivating the response or looming in the background. (...) In particular, I argue that pace Andrew del Rio’s claim in the article “Why Undermining Evolutionary Debunkers is Not Enough,” merely undermining the empirical claim can in fact be a successful line of response to debunkers if one endorses a particular version of doxastic conservatism. I also argue that examining how satisfying a response to an evolutionary debunking argument seems to us and considering what theory of justification is looming in the background of the response can shed light on the plausibility of certain theories of epistemic justification. (shrink)

I investigate the role of geometric intuition in Frege’s early mathematical works and the significance of his view of the role of intuition in geometry to properly understanding the aims of his logicist project. I critically evaluate the interpretations of Mark Wilson, Jamie Tappenden, and Michael Dummett. The final analysis that I provide clarifies the relationship of Frege’s restricted logicist project to dominant trends in German mathematical research, in particular to Weierstrassian arithmetization and to the Riemannian conceptual/geometrical tradition at Göttingen. (...) Concurring with Tappenden, I hold that Frege’s logicism should not be understood as a continuing a project of reductionist arithmetization. However, Frege does not quite take up the Riemannian banner either. His logicism supports a hierarchical understanding of the structure of mathematical knowledge, according to which arithmetic is applicable to geometry but not vice versa because the former is more general, as revealed by the strictly logical nature of its objects in comparison to the intuitional nature of geometric objects. I suggest, in particular, that Frege intended that foundational work would show the use of geometric intuition in complex analysis, a source of error for Riemann that Weierstrass was proud to have uncovered, to be inessential. (shrink)

Imre Lakatos' views on the philosophy of mathematics are important and they have often been underappreciated. The most obvious lacuna in this respect is the lack of detailed discussion and analysis of his 1976a paper and its implications for the methodology of mathematics, particularly its implications with respect to argumentation and the matter of how truths are established in mathematics. The most important themes that run through his work on the philosophy of mathematics and which culminate in the 1976a paper (...) are (1) the (quasi-)empirical character of mathematics and (2) the rejection of axiomatic deductivism as the basis of mathematical knowledge. In this paper Lakatos' later views on the quasi-empirical nature of mathematical theories and methodology are examined and specific attention is paid to what this view implies about the nature of mathematical argumentation and its relation to the empirical sciences. (shrink)

In this paper I investigate two notions of concepts that have played a dominant role in 20th century philosophy of mathematics. According to the first, concepts are definite and fixed; in contrast, according to the second notion concepts are open and subject to modifications. The motivations behind these two incompatible notions and how they can be used to account for conceptual change are presented and discussed. On the basis of historical developments in mathematics I argue that both notions of concepts (...) capture important aspects of mathematical and scientific reasoning, and, consequently, that a pluralistic approach that allows for representing both of these aspects is most useful for an adequate account of investigative practices. (shrink)

One of Adrian Piper’s “reactive guerrilla performances” dealing with issues of race and racism was a calling card that she handed out to individuals who made racist remarks that they would not have made if they had taken themselves to be in the presence of a person of color. The card reads.