Leo Corry tells the story behind the idea of number, from the early days of the Pythagoreans, up until the turn of the twentieth century. He presents an overview of how numbers were handled and conceived in classical Greek mathematics, in the mathematics of Islam, in European mathematics of the middle ages and the Renaissance, during the scientific revolution, all the way through to the mathematics of the 18th to the early 20th century.

This paper takes up Huw Price׳s challenge to develop a retrocausal toy model of the Bell-EPR experiment. I develop three such models which show that a consistent, local, hidden-variables interpretation of the EPR experiment is indeed possible, and which give a feel for the kind of retrocausation involved. The first of the models also makes clear a problematic feature of retrocausation: it seems that we cannot interpret the hidden elements of reality in a retrocausal model as possessing determinate dispositions to (...) affect the outcome of experiments. This is a feature which Price has embraced, but Gordon Belot has argued that this feature renders retrocausal interpretations “unsuitable for formal development”, and the lack of such determinate dispositions threatens to undermine the motivation for hidden-variables interpretations in the first place. But Price and Belot are both too quick in their assessment. I show that determinate dispositions are indeed consistent with retrocausation. What is more, I show that the ontological economy allowed by retrocausation holds out the promise of a classical understanding of spin and polarization. (shrink)

Jacques Derrida’s and Gilles Deleuze’s philosophies of justice share many similar features. For both, justice involves an overturning of law by extralegal means, made possible by an “undecidability” in the judgment-making process. To distinguish their conceptions of justice, we examine their implicit modes of non-classical reasoning with regard to “otherness,” building from Routley and Routley and Daniel Smith, to conclude that Derrida’s thinking on justice is at least paracomplete (or analetheic) while Deleuze’s is just paraconsistent (or dialetheic).

The ArgumentRecent studies in the philosophy of mathematics have increasingly stressed the social and historical dimensions of mathematical practice. Although this new emphasis has fathered interesting new perspectives, it has also blurred the distinction between mathematics and other scientific fields. This distinction can be clarified by examining the special interaction of thebodyandimagesof mathematics.Mathematics has an objective, ever-expanding hard core, the growth of which is conditioned by socially and historically determined images of mathematics. Mathematics also has reflexive capacities unlike those of (...) any other exact science. In no other exact science can the standard methodological framework usedwithinthe discipline also be used to study the nature of the discipline itself.Although it has always been present in mathematical research, reflexive thinking has become increasingly central to mathematics over the past century. Many of the images of the discipline have been dictated by the increase in reflexive thinking which has also determined a great portion of the contemporary philosophy and historiography of mathematics. (shrink)

Connecting several strands of Aristotle's thought, Zoli Filotas sheds light on one of the axioms of Aristotle's ethics and political philosophy - that every community has a ruler - and demonstrates its relevance to his ideas on personal relationships. Aristotle and the Ethics of Difference, Friendship, and Equality reveals a pluralistic theory of rule in Aristotle's thought, tracing it through his corpus and situating it in a discussion among such figures as Gorgias, Xenophon, and Plato. Considering the similarities and (...) differences among various forms of rule, Filotas shows that for Aristotle even virtuous friends must exercise a version of rule akin to that of slaveholders. He also explores why Aristotle distinguishes the hierarchical rule over women from both the mastery of slaves and the political rule exercised by free and equal citizens. In doing so, he argues that natural and social differences among human beings play a complex, and troubling, role in Aristotle's reasoning. Illuminating and thought-provoking, this book reveals Aristotle's ambivalence about political relations and the equal treatment they involve and offers an engaging inquiry into how he understood the common structures of human relationships. (shrink)

Enactivism and ecological psychology converge on the relevance of the environment in understanding perception and action. On both views, perceiving organisms are not merely passive receivers of environmental stimuli, but rather form a dynamic relationship with their environments in such a way that shapes how they interact with the world. In this paper, I suggest that while enactivism and ecological psychology enjoy a shared specification of the environment as the cognitive domain, on both accounts, the structure of the environment, itself, (...) is unspecified beyond that of contingent relations with the species-typical sensorimotor capacities of perceiving organisms. This lack of specification creates a considerable gap in theory regarding the organization of organisms as coupled with their environments. I argue that this gap can be filled by drawing from resources in developmental systems theory, namely, specifying the environmental state-space as a developmental niche that shapes and is shaped by individual organisms over developmental and, on a population scale, evolutionary time. Defining the environment as an organism’s developmental niche makes it clearer how and why certain contingencies have arisen, in turn, strengthening a joint appeal to both enactivism and ecological psychology as theories asserting complementarity between organisms and their environments. (shrink)

For over 20 years, Jaegwon Kim’s Causal Exclusion Argument has stood as the major hurdle for non-reductive physicalism. If successful, Kim’s argument would show that the high-level properties posited by non-reductive physicalists must either be identical with lower-level physical properties, or else must be causally inert. The most prominent objection to the Causal Exclusion Argument—the so-called Overdetermination Objection—points out that there are some notions of causation that are left untouched by the argument. If causation is simply counterfactual dependence, for example, (...) then the Causal Exclusion Argument fails. Thus, much of the existing debate turns on the issue of which account of causation is appropriate. In this paper, however, I take a bolder approach and argue that Kim’s preferred version of the Causal Exclusion Argument fails no matter what account one gives of causation. Any notion of causation that is strong enough to support the premises of the argument is too strong to play the role required in the logic of the argument. I also consider a second version of the Causal Exclusion Argument, and suggest that although it may avoid the problems of the first version, it begs the question against a particular form of non-reductive physicalism, namely emergentism. (shrink)

The world is a complex place, and this complexity is an obstacle to our attempts to explain, predict, and control it. In Power and Influence, Richard Corry investigates the assumptions that are built into the reductive method of explanation—the method whereby we deal with complexity by studying the components of a complex system in relative isolation and use the information so gained to explain or predict the behaviour of the complex whole. He investigates the metaphysical presuppositions built into the reductive (...) method, seeking to ascertain what the world must be like in order that the method could work. Corry argues that the method assumes the existence of causal powers that manifest causal influence—a relatively unrecognised ontological category, of which forces are a paradigm example. The success of the reductive method, therefore, is an argument for the existence of such causal influences. The book goes on to show that adding causal influence to our ontology gives us the resources to solve some traditional problems in the metaphysics of causal powers, laws of nature, causation, emergence, and possibly even normative ethics. What results, then, is not just an understanding of the reductive method, but an integrated metaphysical worldview that is grounded in an ontology of power and influence. (shrink)

One of the most powerful tools in science is the analytic method, whereby we seek to understand complex systems by studying simpler sub-systems from which the complex is composed. If this method is to be successful, something about the sub-systems must remain invariant as we move from the relatively isolated conditions in which we study them, to the complex conditions in which we want to put our knowledge to use. This paper asks what this invariant could be. The paper shows (...) that the kinds of thing that a Humean might point to – behaviour, laws, and dispositions – cannot play the role required of the invariant in question. Nor, indeed, can non-Humean causal powers of the kind advocated by contemporary metaphysicians such as Ellis and Lierse. The paper suggests that the analytic method presupposes a kind of entity that does not appear in standard ontologies – a metaphysically substantial notion of causal influence. This notion of causal influence is one that Cartwright has also seen the need for, though she does not seem to take the notion as seriously as she should. (shrink)

The typical college lecture hall is a highly artificial environment: windowless, fluorescent-lit, and technology-heavy. It all but necessitates treating students as mental receptacles, where learning is a matter of passive absorption of knowledge, and where it is increasingly difficult to hold students’ attention. Natural environments, such as forests and public parks, provide a striking comparison—they free us from technological distractions, invigorate our senses, and encourage physical in addition to mental exploration. What’s more, research in environmental psychology suggests that natural environments (...) restore our ability to focus and pay attention. This article investigates how natural environments might support successful learning in virtue of how such environments can restore our attentional capacities. (shrink)

A process philosophical interpretation of Deleuze's theories of time encounters problems when formulating an account of Deleuze's portrayal of temporality in The Time-Image, where time is understood as having the structure of instantaneity and simultaneity. I remedy this shortcoming of process philosophical readings by formulating a phenomenological interpretation of Deleuze's second synthesis of time. By employing Deleuze's logic of affirmative synthetic disjunction in combination with his differential calculus interpretation of Spinoza's and Bergson's duration, this phenomenological interpretation portrays time as given (...) to our awareness in immediacy rather than through a continuous process of unfolding. The viability of this alternate approach calls into question the claims that Deleuze is strictly a process philosopher and anti-phenomenologist. (shrink)

By means of Vivian Sobchack’s semiotic film phenomenology, we may examine our immediate perceptual acts in film experience in order to determine the ways that the primordial language of embodied existence found at this primary level grounds the secondary level of the more explicit interpretations we give to the film’s elements. Although Gilles Deleuze is openly defiant toward the phenomenological tradition, his studies of film experience can serve this purpose as well, because he is interested in the direct and pre-verbal (...) significance of cinematic images. To bring his observations more fruitfully into film phenomenological studies, I will examine his notion of the discordantly operating body and offer a phenomenological interpretation for his notion of cinematic signs. I then apply this Deleuzian semiotic film phenomenology to his analysis of deep focus cinematography in Orson Welles’ Citizen Kane. When watching one particular scene, different layers of our film experience, namely, those of visual and of temporal depth, collide in such a way that they provide the phenomenal basis for us to produce a temporal interpretation of the spatial relations held between the displayed images. (shrink)

This article explores the changing relationships between geometric and arithmetic ideas in medieval Europe mathematics, as reflected via the propositions of Book II of Euclid’s Elements. Of particular interest is the way in which some medieval treatises organically incorporated into the body of arithmetic results that were formulated in Book II and originally conceived in a purely geometric context. Eventually, in the Campanus version of the Elements these results were reincorporated into the arithmetic books of the Euclidean treatise. Thus, while (...) most of the Latin versions of the Elements had duly preserved the purely geometric spirit of Euclid’s original, the specific text that played the most prominent role in the initial passage of the Elements from manuscript to print—i.e., Campanus’ version—followed a different approach. On the one hand, Book II itself continued to appear there as a purely geometric text. On the other hand, the first ten results of Book II could now be seen also as possiblytranslatable into arithmetic, and in many cases even as inseparably associated with their arithmetic representation. (shrink)

A dispositional property is a tendency, or potency, to manifest some characteristic behaviour in some appropriate context. The mainstream view in the twentieth century was that such properties are to be explained in terms of more fundamental non-dispositional properties, together with the laws of nature. In the last few decades, however, a rival view has become popular, according to which some properties are essentially dispositional in nature, and the laws of nature are to be explained in terms of these fundamental (...) dispositions. The supposed ability of fundamental dispositions to ground natural laws is one of the most attractive features of the dispositional essentialist position. In this paper, however, I cast doubt on the ability of dispositional essences to ground the laws of nature. In particular I argue that the dispositional essentialist position is not able to coherently respond—sympathetically or otherwise—to Cartwright's challenge that there are no true general laws of nature. (shrink)

Jean-Pierre Dupuy argues that our failure to prevent the looming climate catastrophe results from a faulty metaphysics of time: because we believe the present can proceed down one of the many branches that extend into the future, some of which bypass the catastrophe, we do not think it is absolutely urgent to take drastic action now. His solution to this problem of demotivation is “enlightened doomsaying” in “projected time”, which means that we affirm the coming catastrophe as something real in (...) the future rather than being a mere possibility; thus, we regard it seriously enough that we are motivated to take the needed actions to prevent it. One potential obstacle to this proposal is that it requires the forming of consensus and coordination with the powerful players who benefit from our current path and whose apparently near-total grip on this catastrophic future may itself discourage action. We then consider an alternative model based on Gilles Deleuze’s philosophy of the present–future relation. Although it has the branching structure that Dupuy is wary of, it may not suffer from the same problem of demotivation on account of the way it conceives the complex structure of the present event. For this reason, the Deleuzian model may be more suited to motivating action in a world where the future must be fought for rather than unanimously agreed upon. (shrink)

This paper proposes a revision of our understanding of causation that is designed to address what Hartry Field has suggested is the central problem in the metaphysics of causation today: reconciling Bertrand Russell’s arguments that the concept of causation can play no role in the advanced sciences with Nancy Cartwright’s arguments that causal concepts are essential to a scientific understanding of the world. The paper shows that Russell’s main argument is, ironically, very similar to an argument that Cartwright has put (...) forward against the truth of universal laws of nature. The paper uses this insight to develop an account of causation that does justice to traditional views yet avoids the arguments of Russell. (shrink)

One of the most powerful tools in science is the analytic method, whereby we seek to understand complex systems by studying simpler sub-systems from which the complex is composed. If this method is to be successful, something about the sub-systems must remain invariant as we move from the relatively isolated conditions in which we study them, to the complex conditions in which we want to put our knowledge to use. This paper asks what this invariant could be. The paper shows (...) that the kinds of thing that a Humean might point to – behaviour, laws, and dispositions – cannot play the role required of the invariant in question. Nor, indeed, can non-Humean causal powers of the kind advocated by contemporary metaphysicians such as Ellis and Lierse. The paper suggests that the analytic method presupposes a kind of entity that does not appear in standard ontologies – a metaphysically substantial notion of causal influence. This notion of causal influence is one that Cartwright has also seen the need for, though she does not seem to take the notion as seriously as she should. (shrink)

The present article discusses the computational tools (both conceptual and material) used in various attempts to deal with individual cases of FLT, as well as the changing historical contexts in which these tools were developed and used, and affected research. It also explores the changing conceptions about the role of computations within the overall disciplinary picture of number theory, how they influenced research on the theorem, and the kinds of general insights thus achieved. After an overview of Kummer’s contributions and (...) its immediate influence, I present work that favored intensive computations of particular cases of FLT as a legitimate, fruitful, and worth-pursuing number-theoretical endeavor, and that were part of a coherent and active, but essentially low-profile tradition within nineteenth century number theory. This work was related to table making activity that was encouraged by institutions and individuals whose motivations came mainly from applied mathematics, astronomy, and engineering, and seldom from number theory proper. A main section of the article is devoted to the fruitful collaboration between Harry S. Vandiver and Emma and Dick Lehmer. I show how their early work led to the hesitant introduction of electronic computers for research related with FLT. Their joint work became a milestone for computer-assisted activity in number theory at large. (shrink)

In the present article two possible meanings of the term mathematical structure are discussed: a formal and a nonformal one. It is claimed that contemporary mathematics is structural only in the nonformal sense of the term. Bourbaki's definition of structure is presented as one among several attempts to elucidate the meaning of that nonformal idea by developing a formal theory which allegedly accounts for it. It is shown that Bourbaki's concept of structure was, from a mathematical point of view, a (...) superfluous undertaking. This is done by analyzing the role played by the concept, in the first place, within Bourbaki's own mathematical output. Likewise, the interaction between Bourbaki's work and the first stages of category theory is analyzed, on the basis of both published texts and personal documents. (shrink)

Public participation in scientific research has gained prominence in many scientific fields, but the theory of participatory research is still limited. In this paper, we suggest that the divergence of values and goals between academic researchers and public participants in research is key to analyzing the different forms this research takes. We examine two existing characterizations of participatory research: one in terms of public participants' role in the research, the other in terms of the virtues of the research. In our (...) view, each of these captures an important feature of participatory research but is, on its own, limited in what features it takes into account. We introduce an expanded conception of norms of collaboration that extends to both academic researchers and public participants. We suggest that satisfying these norms requires consideration of the two groups' possibly divergent values and goals, and that a broad characterization of participatory research that starts from participants' values and goals can motivate both public participants’ role in the research and the virtues of the research. The resulting framework clarifies the similarities and differences among participatory projects and can help guide the responsible design of such projects. (shrink)

The history of Israeli science and technology offers a unique case study of a young and small nation that has developed an unprecedented love affair with science and technology. Unlike other nineteenth-century ideologies, Zionism was never considered to be founded on science. Nevertheless, from the very start, the Zionist movement perceived the sciences, pure and applied, as central to its program of creating a new Jewish society in the Land of Israel. Modern science was to provide twice for the Jews: (...) a relief from their suffocating religion and the tools needed to recover their ancient land from its ruins. Israel would remain the people of the book, but it would be the Book of Nature, not of God, that would set it free. Sharing the universal knowledge and values of science with mankind, the Jews would finally become both normal and self-determined. Thus, already in the nineteenth century, long before the State of Israel was founded, Zionist visionaries had dreamt of it as a modern version of Francis Bacon's utopian Kingdom of Bensalem, where science and technology would provide health, wealth, and power. (shrink)

‘Intercultural Theology’ began in the 1970’s as a way of expressing theological mutuality and equality between cultures. Since then, the word ‘intercultural’ has gained currency in a number of fields, secular and religious. This article explores the possibilities of speaking of mission as ‘intercultural’ rather than ‘cross-cultural’ as a way of expressing the cultural engagement that is more appropriate to a postmodern world. The inadequacies of the cross-cultural approach, with its roots in modernity, are examined in the light of the (...) complexity and diversity of cultural identities today. Intercultural mission in contrast is relational, mutual, dialogical, open-ended, and creates space within which God’s missio dei can be experienced. Two examples are offered of intercultural mission in practice. It is suggested that this resonates with Bosch’s proposal for a new paradigm for mission in a postmodern context. (shrink)

We mourn the loss of Yehuda Elkana, founding editor of this journal. Setting science in context was a mission of his life. For him this did not mean to relativize and historicize science to the point where it is no longer distinguishable as central to the human quest for knowledge. Rather, an understanding of science as being rooted in social, material, and cultural contexts was for him the key to its central role for solving the problems of humanity with which (...) he was so deeply concerned. (shrink)

Open peer commentary on the article “Beyond Individual-Centred 4E Cognition: Systems Biology and Sympoiesis” by Mads Julian Dengsø & Michael David Kirchhoff. Abstract: A move to sympoiesis entails a conceptual overhaul of autopoiesis, a key facet of the enactive approach to cognition. In this commentary, I invite more systematic thinking about the concept of autopoiesis and enactivist commitments to it, given the inconsistencies regarding individuality raised in the target article.

French philosopher Gilles Deleuze wrote two 'logic' books: Francis Bacon: The Logic of Sensation and The Logic of Sense. However, in neither of these books nor in any other works does Deleuze articulate in a formal way the features of the logic he employs. He certainly does not use classical logic. And the best options for the non-classical logic that he may be implementing are: fuzzy, intuitionist, and many-valued. These are applicable to his concepts of heterogeneous composition and becoming, affirmative (...) synthetic disjunction, and powers of the false. In The Logic of Gilles Deleuze. Volume 1: Basic Principles, Corry Shores examines the applicability of three non-classical logics to Deleuze's philosophy, by building from the philosophical and logical writings of Graham Priest, the world's leading proponent of dialetheism. Through so doing, Shores demonstrates how Deleuze's logic is best understood as a dialetheic, paraconsistent, many-valued logic. (shrink)

The continental philosophy of non-classical logics is a relatively new field that seeks to determine whether any aspects of certain continental philosophers’ thinking can be characterized in terms of non-classical logics. Some of the main figures that have been examined so far are Martin Heidegger, Jacques Derrida, Gilles Deleuze, and François Laruelle. Although many of these studies are grounded in the writings of Graham Priest, who wrote some of the seminal texts in the field, Jc Beall’s work also features prominently (...) in a number of cases. After surveying this field and highlighting Beall’s influence within it, I claim that it can be substantially expanded by making further uses of Beall’s writings, especially ones that introduce non-classical logics in ways that are especially suitable for this particular field of study and also that discuss such topics as negation, subclassical logics, logical pluralism, and glutty futures. (shrink)

The difference between cause and effect seems obvious and crucial in ordinary life, yet missing from modern physics. Almost a century ago, Bertrand Russell called the law of causality 'a relic of a bygone age'. In this important collection 13 leading scholars revisit Russell's revolutionary conclusion, discussing one of the most significant and puzzling issues in contemporary thought.

Typically, we think of the brain as responsible for cognition. But the brain is, importantly, embedded in a body—a body that moves around and interacts with features of the environment. What role, then, does the body play in cognition? Some philosophers would argue that it has no significant role in determining how we think about cognitive processing. But others argue that the body is fundamental to cognition, because the body is deeply involved with cognitive processes such as acting and perceiving. (...) Embodied and enactive cognition are theories that emphasize the importance of the body in understanding cognition. This chapter discusses how some philosophers and cognitive scientists have investigated bodily involvement in cognition. -/- We begin in Section 1 by providing an overview of the historical developments that influenced enactive and embodied cognitive science, namely, phenomenology, ecological psychology, and the phenomenological critique of artificial intelligence. We show how these developments were advanced by the introduction of dynamical systems theorizing in psychology and situatedness in robotics. Embodied, extended cognition and enactivism are then introduced as theories that emerge from this discussion. In Section 2, we consider some key debates in embodiment and enactivism, including those over the role of representations and the status of dynamical modeling. We end in Section 3 by highlighting some of the innovative research developments that have come out of an embodied approach to cognition, with a focus on insights regarding social cognition. (shrink)

To compare Merleau-Ponty’s and Deleuze’s phenomenal bodies, I first examine how for Merleau-Ponty phenomena appear on the basis of three levels of integration: 1) between the parts of the world, 2) between the parts of the body, and 3) between the body and its world. I contest that Deleuze’s attacks on phenomenology can be seen as constructive critiques rather than as being expressions of an anti-phenomenological position. By building from Deleuze’s definition of the phenomenon and from his more phenomenologically relevant (...) writings, we find that phenomena for him are given to the body under exactly the opposite conditions as for Merleau-Ponty, namely that 1) the world’s differences 2) appear to a disordered body that 3) comes into shocking affective contact with its surroundings. I argue that a Deleuzian theory of bodily-given phenomena is better suited than Merleau-Ponty’s model in the task of accounting for the intensity of phenomenal appearings. (shrink)