Over the history of the study of visual perception there has been great success at discovering countless visual illusions. There has been less success in organizing the overwhelming variety of illusions into empirical generalizations (much less explaining them all via a unifying theory). Here, this article shows that it is possible to systematically organize more than 50 kinds of illusion into a 7 × 4 matrix of 28 classes. In particular, this article demonstrates that (1) smaller sizes, (2) slower speeds, (...) (3) greater luminance contrast, (4) farther distance, (5) lower eccentricity, (6) greater proximity to the vanishing point, and (7) greater proximity to the focus of expansion all tend to have similar perceptual effects, namely, to (A) increase perceived size, (B) increase perceived speed, (C) decrease perceived luminance contrast, and (D) decrease perceived distance. The detection of these empirical regularities was motivated by a hypothesis, called “perceiving the present,” that the visual system possesses mechanisms for compensating neural delay during forward motion. This article shows how this hypothesis predicts the empirical regularity. (shrink)

In “The Responsibilities and Role of Business in Relation to Society,” Nien-hê Hsieh challenges Joseph Heath’s “market failure” or Paretian approach to business ethics by arguing for a “Back to Basics” approach. Here, I argue that two basics of Hsieh’s three-basics vision are flawed, because a. ordinary morality is in fact not sufficient for the adversarial realm of the market, and b. the ideal of a Pareto-optimal market economy with perfect competition does in fact provide an adequate basis (...) for normative rules against market failures. (shrink)

One of the key achievements of critical realism has been to expose the modernist myth of universal reason, which holds that authentic knowledge claims must be objectively ‘pure’, uncontaminated by the subjectivity of local place, specific time and particular culture. Wright aims to address the lack of any substantial and sustained engagement between critical realism and theological critical realism with particular regard to: (a) the distinctive ontological claims of Christianity; (b) their epistemic warrant and intellectual legitimacy; and (c) scrutiny of (...) the primary source of the ontological claims of Christianity, namely the historical figure of Jesus of Nazareth. As such, it functions as a prolegomena to a much needed wider debate, guided by the under-labouring services of critical realism, between Christianity and various other religious and secular worldviews. This important new text will help stimulate a debate that has yet to get out of first gear. This book will appeal to academics, graduate and post-graduate students especially, but also Christian clergy, ministers and informed laity, and members of the general public concerned with the nature of religion and its place in contemporary society. (shrink)

Against a view currently popular in the literature, it is argued that Kant was not a niıve realist about perceptual experience. Naive realism entails that perceptual experience is object-dependent in a very strong sense. In the first half of the paper, I explain what this claim amounts to and I undermine the evidence that has been marshalled in support of attributing it to Kant. In the second half of the paper, I explore in some detail Kant’s account of hallucination and (...) argue that no such account is available to someone who thinks that veridical perceptual experience is object-dependent in the naive realist sense. Kant’s theory provides for a remarkably sophisticated, bottom-up explanation of the phenomenal character of hallucinatory episodes and is crucial for gaining a proper understanding of his model of the mind and its place in nature. (shrink)

Baghdad, California -- Calm down -- Clearing the dust -- Proof -- The big bang -- Evolution -- Evil -- Religion -- A crisis of reason.

Critical Realism and Marxism addresses controversial debates, revealing a potentially fruitful relationship; deepening our understanding of the social world and contibuting towards eliminating barbarism in contemporary capitalism.

Mereological nihilism (henceforth just "nihilism") is the thesis that composition never occurs. Nihilism has often been defended on the basis of its theoretical simplicity, including its ontological simplicity and its ideological simplicity (roughly, nihilism's ability to do without primitive mereological predicates). In this paper I defend nihilism on the basis of the theoretical unification conferred by nihilism, which is, roughly, nihilism's capacity to allow us to take fewer phenomena as brute and inexplicable. This represents a respect in which nihilism enjoys (...) greater theoretical simplicity than its rivals which has not yet been explored, and which is immune to many of the objections which have been leveled against previous arguments for nihilism from nihilism's theoretical simplicity. Composition as identity might be thought to confer a similar degree of theoretical unification as nihilism. I end the paper by arguing that this is not the case. (shrink)

This essay considers Kant’s theory of modality in light of a debate in contemporary modal metaphysics and modal logic concerning the Barcan formulas. The comparison provides a new and fruitful perspective on Kant’s complex and sometimes confusing claims about possibility and necessity. Two central Kantian principles provide the starting point for the comparison: that the possible must be grounded in the actual and that existence is not a real predicate. Both are shown to be intimately connected to the Barcan formulas, (...) and Kant’s views on what he distinguishes as three different kinds of modality are then considered in light of this connection. (shrink)

ABSTRACT:The articles in the special issue ofBusiness Ethics Quarterly, “Normative Business Ethics in a Global Economy: New Directions on Donaldsonian Themes,” were written by a set of outstanding scholars: Margaret M. Blair, Joseph P. Gaspar, Nien-hê Hsieh, Peter L. Jennings, Marietta Peytcheva, Andreas Georg Scherer, Amy J. Sepinwall, Andrew Stark, Danielle E. Warren, and Manuel Velasquez. In this commentary I reply to my colleagues, arranging my reply around the following themes: 1) the corporate moral agent; 2) the idea (...) of a social contract for business; 3) managing ethics within corporations; and 4) values in business. I discuss each in turn. However, I reflect first on my idiosyncratic approach to business ethics. (shrink)

Kris McDaniel has recently proposed an interpretation of the distinction between conventional truth and ultimate truth, as that distinction is made within Abhidharma metaphysics. According to McDaniel's proposal, the distinction between conventional truth and ultimate truth is closely connected with a similar distinction between conventional existence and ultimate existence. What is more, the distinction between conventional existence and ultimate existence should be interpreted along ontological pluralist lines: the difference between things that ultimately exist and things that merely conventionally exist amounts (...) to a difference in the modes of being enjoyed by the things in question.One... (shrink)

Sand drawings are introduced in relation to the fieldwork of British anthropologists John Layard and Bernard Deacon early in the twentieth century, and the status of sand drawings as mathematics is discussed in the light of Wittgenstein’s idea that “in mathematics process and result are equivalent”. Included are photographs of the illustrations in Layard’s own copy of Deacon’s “Geometrical Drawings from Malekula and other Islands of the New Hebrides” (1934). This is a brief companion to my article “Wittgenstein on string (...) figures as mathematics” (2022), published in the journal Philosophical Investigations. (shrink)

An increasing amount of contemporary philosophy of mathematics posits, and theorizes in terms of special kinds of mathematical modality. The goal of this paper is to bring recent work on higher-order metaphysics to bear on the investigation of these modalities. The main focus of the paper will be views that posit mathematical contingency or indeterminacy about statements that concern the `width' of the set theoretic universe, such as Cantor's continuum hypothesis. Within a higher-order framework I show that contingency about the (...) width of the set-theoretic universe refutes two orthodoxies concerning the structure of modal reality: the view that the broadest necessity has a logic of S5, and the "Leibniz biconditionals" stating that what is possible, in the broadest sense of possible, is what is true in some possible world. Nonetheless, I suggest that the underlying picture of modal set-theory is coherent and has attractions. (shrink)

A list of resources for virtue theories of argumentation. Last updated October 31st, 2023.

It has been a decade since the phrase virtue argumentation was introduced, and while it would be an exaggeration to say that it burst onto the scene, it would be just as much of an understatement to say that it has gone unnoticed. Trying to strike the virtuous mean between the extremes of hyperbole and litotes, then, we can fairly characterize it as a way of thinking about arguments and argumentation that has steadily attracted more and more attention from argumentation (...) theorists. We hope it is neither too late for an introduction to the field nor too soon for some retrospective assessment of where things stand. (shrink)

“Doğal din” terimi, bazen doğanın kendisinin ilahi olduğu bir panteistik doktrine atıfta bulunur. “Doğal teoloji” terimi ise aksine, başlangıçta gözlemlenen doğal gerçekler temelinde (ve bazen) Tanrı’nın varlığını savunmaya yönelik projeye atıfta bulunur. Bununla birlikte çağdaş felsefede, hem “doğal din” hem de “doğal teoloji” genel olarak, dinî veya teolojik konuları araştırmak için insana, “doğal” olan bilişsel yetilerini – akıl, algı, içgözlem- kullanma projesini ifade eder. Doğal din veya teoloji, mevcut anlayış üzerine, doğayla ilgili ampirik araştırmalarla sınırlı olmamakla birlikte ayrıca panteistik bir (...) sonuçla da bağlantılı değildir. Bununla birlikte o, özel doğal olmayan yetilere (ESP, telepati, mistik deneyim) veya doğaüstü bilgi kaynaklarına (kutsal metinler, vahiy teolojisi, inanç otoriteleri, doğrudan doğaüstü iletişim) hitap etmekten kaçınır. Genel olarak, doğal din veya teoloji (bundan böyle “doğal teoloji” olarak anılacaktır), diğer felsefi ve bilimsel girişimlerle aynı rasyonel inceleme standartlarına uymayı amaçlamaktadır ve aynı değerlendirme ve eleştiri yöntemlerine tabidir. (shrink)

The traditional view of evidence in mathematics is that evidence is just proof and proof is just derivation. There are good reasons for thinking that this view should be rejected: it misrepresents both historical and current mathematical practice. Nonetheless, evidence, proof, and derivation are closely intertwined. This paper seeks to tease these concepts apart. It emphasizes the role of argumentation as a context shared by evidence, proofs, and derivations. The utility of argumentation theory, in general, and argumentation schemes, in particular, (...) as a methodology for the study of mathematical practice is thereby demonstrated. Argumentation schemes represent an almost untapped resource for mathematics education. Notably, they provide a consistent treatment of rigorous and non-rigorous argumentation, thereby working to exhibit the continuity of reasoning in mathematics with reasoning in other areas. Moreover, since argumentation schemes are a comparatively mature methodology, there is a substantial body of existing work to draw upon, including some increasingly sophisticated software tools. Such tools have significant potential for the analysis and evaluation of mathematical argumentation. The first four sections of the paper address the relationships of evidence to proof, proof to derivation, argument to proof, and argument to evidence, respectively. The final section directly addresses some of the educational implications of an argumentation scheme account of mathematical reasoning. (shrink)

A Physicalist Manifesto is a full treatment of the comprehensive physicalist view that, in some important sense, everything is physical. Andrew Melnyk argues that the view is best formulated by appeal to a carefully worked-out notion of realization, rather than supervenience; that, so formulated, physicalism must be importantly reductionist; that it need not repudiate causal and explanatory claims framed in non-physical language; and that it has the a posteriori epistemic status of a broad-scope scientific hypothesis. Two concluding chapters argue (...) in detail that contemporary science provides no significant empirical evidence against physicalism and some considerable evidence for it. Written in a brisk, candid and exceptionally clear style, this 2003 book should appeal to professionals and students in philosophy of mind, metaphysics and philosophy of science. (shrink)

In their account of theory change in logic, Aberdein and Read distinguish 'glorious' from 'inglorious' revolutions--only the former preserves all 'the key components of a theory' [1]. A widespread view, expressed in these terms, is that empirical science characteristically exhibits inglorious revolutions but that revolutions in mathematics are at most glorious [2]. Here are three possible responses: 0. Accept that empirical science and mathematics are methodologically discontinuous; 1. Argue that mathematics can exhibit inglorious revolutions; 2. Deny that inglorious revolutions are (...) characteristic of science. Where Aberdein and Read take option 1, option 2 is preferred by Mizrahi [3]. This paper seeks to resolve this disagreement through consideration of some putative mathematical revolutions. [1] Andrew Aberdein and Stephen Read, The philosophy of alternative logics, The Development of Modern Logic (Leila Haaparanta, ed.), Oxford University Press, Oxford, 2009, pp. 613-723. [2] Donald Gillies (ed.), Revolutions in Mathematics, Oxford University Press, Oxford, 1992. [3] Moti Mizrahi, Kuhn's incommensurability thesis: What's the argument?, Social Epistemology 29 (2015), no. 4, 361-378. (shrink)

In this paper, I defend what may seem a surprising view: that John Stuart Mill’s famous harm principle would, if taken to be what justifies government action, disallow the existence of corporations. My claim is not that harmful activities of currently existing corporations warrants their losing corporate status according to the harm principle. The claim, rather, is that taken strictly, the harm principle and the legal possibility of incorporation are mutually exclusive. This view may be surprising—and I do not at (...) all mean to attribute it to Mill—but if I am right, it should be obvious. It should also encourage us to think more about the nature of the markets within which business occurs. (shrink)

This chapter explores two kinds of testimonial trust, what we call ‘evidential trust’ and ‘non-evidential trust’ with the aim of asking how testimonial trust could provide epistemic reasons for belief. We argue that neither evidential nor non-evidential trust can play a distinctive role in providing evidential reasons for belief, but we tentatively propose that non-evidential trust can in some circumstances provide a novel kind of epistemic reason for belief, a reason of epistemic facilitation. The chapter begins with an extensive discussion (...) of standard accounts of both kinds of trust and criticises especially the standard accounts of non-evidential trust. A new account of non-evidential trust is offered that avoids a number of difficulties that plague the standard accounts by rejecting what we call ‘attitude-liability assumptions’. (shrink)

Wittgenstein's farcical clash with literary critic F. R. Leavis over the analysis of Empson's poem "Legal Fiction" is well known to devotees of Wittgenstein's life (Ludwig Wittgenstein: Personal Recollections (1981), edited by Rush Rhees, Oxford: Basil Blackwell, 80). Less well known is the value of studying Empson's artistic and intellectual achievement as part of the wider cultural background for the appreciation of Wittgenstein's views and influence, early and late. This talk sketches some diverting byways awaiting further exploration. A recurrent theme (...) is contradiction. (shrink)

I intend to bring recent work applying virtue theory to the study of argument to bear on a much older problem, that of disagreements that resist rational resolution, sometimes termed "deep disagreements". Just as some virtue epistemologists have lately shifted focus onto epistemic vices, I shall argue that a renewed focus on the vices of argument can help to illuminate deep disagreements. In particular, I address the role of arrogance, both as a factor in the diagnosis of deep disagreements and (...) as an obstacle to their mutually acceptable resolution. Arrogant arguers are likely to make any disagreements to which they are party seem deeper than they really are and arrogance impedes the strategies that we might adopt to resolve deep disagreements. As a case in point, since arrogant or otherwise vicious arguers cannot be trusted not to exploit such strategies for untoward ends, any policy for deep disagreement amelioration must require particularly close attention to the vices of argument, lest they be exploited by the unscrupulous. (shrink)

This chapter discusses Kant's 1763 "possibility proof" for the existence of God. I first provide a reconstruction of the proof in its two stages, and then revisit my earlier argument according to which the being the proof delivers threatens to be a Spinozistic-panentheistic God—a being whose properties include the entire spatio-temporal universe—rather than the traditional, ontologically distinct God of biblical monotheism. I go on to evaluate some recent alternative readings that have sought to avoid this result by arguing that the (...) relevant facts about real modality can be ultimately grounded in God’s powers or thoughts – or that Kant just leaves the grounding relations mysterious. I argue that the textual and philosophical costs of each of these alternative readings are formidable. The chapter concludes with a discussion of the fate of the proof in the critical period. Some commentators think that it disappears altogether, or that it is downgraded such that it produces a mere regulative idea of God as the most real being. I suggest that the proof survives but that the mode of assent it licenses towards its conclusion changes from knowledge to a certain kind of Belief (Glaube). (shrink)

Douglas Walton’s multitudinous contributions to the study of argumentation seldom, if ever, directly engage with argumentation in mathematics. Nonetheless, several of the innovations with which he is most closely associated lend themselves to improving our understanding of mathematical arguments. I concentrate on two such innovations: dialogue types (§1) and argumentation schemes (§2). I argue that both devices are much more applicable to mathematical reasoning than may be commonly supposed.

Formal logic has often been seen as uniquely placed to analyze mathematical argumentation. While formal logic is certainly necessary for a complete understanding of mathematical practice, it is not sufficient. Important aspects of mathematical reasoning closely resemble patterns of reasoning in nonmathematical domains. Hence the tools developed to understand informal reasoning, collectively known as argumentation theory, are also applicable to much mathematical argumentation. This chapter investigates some of the details of that application. Consideration is given to the many contrasting meanings (...) of the word “argument”; to some of the specific argumentation-theoretic tools that have been applied to mathematics, notably Toulmin layouts and argumentation schemes; to some of the different ways that argumentation is implicated in mathematical practices; and to the social aspects of mathematical argumentation. (shrink)

In this new collection of essays, Andrew Feenberg argues that conflicts over the design and organization of the technical systems that structure our society shape deep choices for the future. A pioneer in the philosophy of technology, Feenberg demonstrates the continuing vitality of the critical theory of the Frankfurt School. He calls into question the anti-technological stance commonly associated with its theoretical legacy and argues that technology contains potentialities that could be developed as the basis for an alternative form (...) of modern society. Feenberg's critical reflections on the ideas of Jürgen Habermas, Herbert Marcuse, Jean-François Lyotard, and Kitaro Nishida shed new light on the philosophical study of technology and modernity. He contests the prevalent conception of technology as an unstoppable force responsive only to its own internal dynamic and politicizes the discussion of its social and cultural construction. This argument is substantiated in a series of compelling and well-grounded case studies. Through his exploration of science fiction and film, AIDS research, the French experience with the "information superhighway," and the Japanese reception of Western values, he demonstrates how technology, when subjected to public pressure and debate, can incorporate ethical and aesthetic values. (shrink)

Vagueness is the study of concepts that admit borderline cases. The epistemology of vagueness concerns attitudes we should have towards propositions we know to be borderline. On this basis Andrew Bacon develops a new theory of vagueness in which vagueness is fundamentally a property of propositions, explicated in terms of its role in thought.

I want to say something about the sort of arguments that it is possible to lose, and whether losing arguments can be done well. I shall focus on losing philosophical arguments, and I will be talking about arguments in the sense of acts of arguing. This is the sort of act that one can perform on one’s own or with one other person in private. But in either of these cases it is difficult to win—or to lose. So I shall (...) concentrate on arguments with audiences. We may think of winning or losing such arguments in terms of whether the audience is convinced. Of course, this doesn’t necessarily have anything to do with who is in the right. That means that there are two sorts of loser: real losers, who lose the argument deservedly, because they are in the wrong, and mere losers, who lose the argument undeservedly, because they are in the right. Hence there must also be two sorts of winner: real winners, who win the argument deservedly, because they are in the right, and mere winners, who win the argument undeservedly, because they are in the wrong. An optimal outcome for arguments with losers would be if all the losers are real losers. (shrink)

What is the function of cognition? On one influential account, cognition evolved to co-ordinate behaviour with environmental change or complexity. Liberal interpretations of this view ascribe cognition to an extraordinarily broad set of biological systems—even bacteria, which modulate their activity in response to salient external cues, would seem to qualify as cognitive agents. However, equating cognition with adaptive flexibility per se glosses over important distinctions in the way biological organisms deal with environmental complexity. Drawing on contemporary advances in theoretical biology (...) and computational neuroscience, we cash these distinctions out in terms of different kinds of generative models, and the representational and uncertainty-resolving capacities they afford. This analysis leads us to propose a formal criterion for delineating cognition from other, more pervasive forms of adaptive plasticity. On this view, biological cognition is rooted in a particular kind of functional organisation; namely, that which enables the agent to detach from the present and engage in counterfactual inference. (shrink)

In this chapter I argue that intellectual humility is related to argumentation in several distinct but mutually supporting ways. I begin by drawing connections between humility and two topics of long-standing importance to the evaluation of informal arguments: the ad verecundiam fallacy and the principle of charity. I then explore the more explicit role that humility plays in recent work on critical thinking dispositions, deliberative virtues, and virtue theories of argumentation.

The literature on metaphysical explanation contains three widely accepted assumptions. First, that the notion of metaphysical explanation with which philosophers are interested is a notion with which the folk are familiar: it is at least continuous with the folk notion. Second, that metaphysical explanations are true propositions of a certain form that are true, (or false), simpliciter. Third, that it is at least the case that mostly, if x metaphysically explains y, then y does not metaphysically explain x. On the (...) basis of empirical investigations that we pursued, we argue that at least two of these assumptions are false . (shrink)

This paper proposes that virtue theories of argumentation and theories of visual argumentation can be of mutual assistance. An argument that adoption of a virtue approach provides a basis for rejecting the normative independence of visual argumentation is presented and its premisses analysed. This entails an independently valuable clarification of the contrasting normative presuppositions of the various virtue theories of argumentation. A range of different kinds of visual argument are examined, and it is argued that they may all be successfully (...) evaluated within a virtue framework, without invoking any novel virtues. (shrink)

Andrew Cooper presents the first systematic study of Kant's account of natural history. Cooper contends that Kant made a decisive contribution to one of the most explosive and understudied revolutions in the history of science: the addition of time to the frame in which explanations are required, sought, and justified in natural science. Through addressing a wide range of Kant's works, Cooper challenges the claim that Kant's theory of science denies a developmental conception of nature and argues instead that (...) it establishes a method by which natural historians can genuinely dispute historical claims and potentially come to consensus. This method, Cooper argues, can be used to expose serious flaws in Kant's own historical reasoning, including the formation and defence of his racist views. The book will be valuable to philosophers seeking to discern both the power and limitations of Kant's theory of science, and to historians of science working on the fractured landscape of eighteenth-century Newtonianism. (shrink)

This chapter argues that a virtue-theoretic account of argumentation can enhance our understanding of the phenomenon of populism and offer some lines of response. Virtue theories of argumentation emphasize the role of arguers in the conduct and evaluation of arguments and lay particular stress on arguers’ acquired dispositions of character, otherwise known as intellectual virtues and vices. One variety of argumentation of particular relevance to democratic decision-making is group deliberation. There are both theoretical and empirical reasons for maintaining that intellectual (...) humility is pivotal to the practice of group deliberation. Several factors to which the rise of populism has been attributed may be understood as failures of the virtues of argumentation in general, and intellectual humility in particular. (shrink)

Heidegger's later thought is a thinking of things, so argues Andrew J. Mitchell in The Fourfold. Heidegger understands these things in terms of what he names "the fourfold"--a convergence of relationships bringing together the earth, the sky, divinities, and mortals--and Mitchell's book is the first detailed exegesis of this neglected aspect of Heidegger's later thought. As such it provides entré to the full landscape of Heidegger's postwar thinking, offering striking new interpretations of the atomic bomb, technology, plants, animals, weather, (...) time, language, the holy, mortality, dwelling, and more. What results is a conception of things as ecstatic, relational, singular, and, most provocatively, as intrinsically tied to their own technological commodification. A major new work that resonates beyond the confines of Heidegger scholarship, The Fourfold proposes nothing less than a new phenomenological thinking of relationality and mediation for understanding the things around us. (shrink)

There has been little overt discussion of the experimental philosophy of logic or mathematics. So it may be tempting to assume that application of the methods of experimental philosophy to these areas is impractical or unavailing. This assumption is undercut by three trends in recent research: a renewed interest in historical antecedents of experimental philosophy in philosophical logic; a “practice turn” in the philosophies of mathematics and logic; and philosophical interest in a substantial body of work in adjacent disciplines, such (...) as the psychology of reasoning and mathematics education. This introduction offers a snapshot of each trend and addresses how they intersect with some of the standard criticisms of experimental philosophy. It also briefly summarizes the specific contribution of the other chapters of this book. (shrink)

America's supposed moral decline from an imagined golden age, and the threat of divine punishment for the sin of straying from the path of righteousness, have been consistent themes in its political and religious rhetoric. In Prodigal Nation, Andrew Murphy investigates the jeremiad's historical roots and probes the ways in which it continues to illuminate themes and tensions in American social and political life.

If good argument is virtuous, then fallacies are vicious. Yet fallacies cannot just be identified with vices, since vices are dispositional properties of agents whereas fallacies are types of argument. Rather, if the normativity of good argumentation is explicable in terms of virtues, we should expect the wrongness of fallacies to be explicable in terms of vices. This approach is defended through case studies of several fallacies, with particular emphasis on the ad hominem.

Normothermic Regional Perfusion (NRP) is increasingly being adopted as an intervention designed to improve organ utilization from donors after circulatory death because of the efficiency with which...

This book explores the results of applying empirical methods to the philosophy of logic and mathematics. Much of the work that has earned experimental philosophy a prominent place in twenty-first century philosophy is concerned with ethics or epistemology. But, as this book shows, empirical methods are just as much at home in logic and the philosophy of mathematics. -/- Chapters demonstrate and discuss the applicability of a wide range of empirical methods including experiments, surveys, interviews, and data-mining. Distinct themes emerge (...) that reflect recent developments in the field, such as issues concerning the logic of conditionals and the role played by visual elements in some mathematical proofs. -/- Featuring leading figures from experimental philosophy and the fields of philosophy of logic and mathematics, this collection reveals that empirical work in these disciplines has been quietly thriving for some time and stresses the importance of collaboration between philosophers and researchers in mathematics education and mathematical cognition. (shrink)