The prime number theorem, established by Hadamard and de la Vallée Poussin independently in 1896, asserts that the density of primes in the positive integers is asymptotic to 1/ln x. Whereas their proofs made serious use of the methods of complex analysis, elementary proofs were provided by Selberg and Erdos in 1948. We describe a formally verified version of Selberg's proof, obtained using the Isabelle proof assistant.

1.1 Some examples of rule induction on permutations . . . . . . . 6 1.2 Ways of making new permutations . . . . . . . . . . . . . . . 7 1.3 Further results . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Removing elements . . . . . . . . . . (...) . . . . . . . . . . . . . . 8.. (shrink)

"This economic system cannot do without the ultima ratio of the complete destruction of those existences which are irretrievably associated with the hopelessly unadapted." "Joseph Schumpeter, The Theory of Economic Development1"Slavoj Žižek poses more than a few heavy-gauge questions in his In Defense of Lost Causes . Foremost among them, at the beginning of chapter nine: “The only true question today is: do we endorse this ‘naturalization’ of capitalism, or does contemporary global capitalism contain antagonisms which are sufficiently strong to (...) prevent its indefinite reproduction?”2 This vast question—as well as its possible answers—develops in Žižek's…. (shrink)

This report highlights and explores five questions which arose from The Unity of Consciousness and Sensory Integration conference at Brown University in November of 2011: 1. What is the relationship between the unity of consciousness and sensory integration? 2. Are some of the basic units of consciousness multimodal? 3. How should we model the unity of consciousness? 4. Is the mechanism of sensory integration spatio-temporal? 5. How Should We Study Experience, Given Unity Relations?

This is an excerpt of a report that highlights and explores five questions which arose from The Unity of Consciousness and Sensory Integration conference at Brown University in November of 2011. This portion of the report explores the question: Is the mechanism of sensory integration spatio-temporal?

This is an excerpt of a report that highlights and explores five questions which arose from The Unity of Consciousness and Sensory Integration conference at Brown University in November of 2011. This portion of the report explores the question: Are some of the basic units of consciousness multimodal?

This is an excerpt of a report that highlights and explores five questions which arose from The Unity of Consciousness and Sensory Integration conference at Brown University in November of 2011. This portion of the report explores the question: How should we model the unity of consciousness?

This is an excerpt of a report that highlights and explores five questions which arose from The Unity of Consciousness and Sensory Integration conference at Brown University in November of 2011. This portion of the report explores the question: How should we study experience, given unity relations?

This is an excerpt from a report that highlights and explores five questions which arose from the workshop on perceptual learning and perceptual recognition at the University of Toronto, Mississauga on May 10th and 11th, 2012. This excerpt explores the question: How do we recognize distinct types of emotion in music?

This is an excerpt from a report that highlights and explores five questions which arose from the workshop on perceptual learning and perceptual recognition at the University of Toronto, Mississauga on May 10th and 11th, 2012. This excerpt explores the question: What counts as cognitive penetration?

This report highlights and explores four questions that arose from the workshop on temporal experience at the University of Toronto, May 20th and 21st, 2013.

This report highlights and explores five questions which arose from the workshop on perceptual learning and perceptual recognition at the University of Toronto, Mississauga on May 10th and 11th, 2012: 1. How should we demarcate perceptual learning from perceptual development? 2. What are the origins of multimodal associations? 3. Does our representation of time provide an amodal framework for multi-sensory integration? 4. What counts as cognitive penetration? 5. How can philosophers and psychologists most fruitfully collaborate?

This is an excerpt from a report that highlights and explores five questions which arose from the workshop on perceptual learning and perceptual recognition at the University of Toronto, Mississauga on May 10th and 11th, 2012. This excerpt explores the question: How should we demarcate perceptual learning from perceptual development?

This is an excerpt from a report that highlights and explores five questions which arose from the workshop on perceptual learning and perceptual recognition at the University of Toronto, Mississauga on May 10th and 11th, 2012. This excerpt explores the question: How can philosophers and psychologists most fruitfully collaborate?

This is an excerpt from a report on the Temporal Experience Workshop at the University of Toronto in May of 2013. This portion of the report explores the question: Do we have one central clock for time, or different clocks for each sense modality?

This is an excerpt from a report on the Temporal Experience Workshop at the University of Toronto in May of 2013. This portion of the report explores the question: What sorts of mechanisms underlie the perceived duration of external events?

This is an excerpt from a report on the Temporal Experience Workshop at the University of Toronto in May of 2013. This portion of the report explores the question: What can we learn about the nature of time from the nature of ordinary experience?

This is an excerpt from a report that highlights and explores five questions which arose from the workshop on perceptual learning and perceptual recognition at the University of Toronto, Mississauga on May 10th and 11th, 2012. This excerpt explores the question: Does our representation of time provide and amodal framework for multi-sensory integration?

This is an excerpt from a report that highlights and explores five questions which arose from the workshop on perceptual learning and perceptual recognition at the University of Toronto, Mississauga on May 10th and 11th, 2012. This excerpt explores the question: What are the origins of multimodal associations?

This is an excerpt from a report on the Temporal Experience Workshop at the University of Toronto in May of 2013. This portion of the report explores the question: What is the relationship between time as represented in experience, the timing of the experiential act, and the timing of the neural realizer of the experience?

This paper delineates the thrust of Augustine's theodicy against the broader background of his Christian Neoplatonic outlook. We examine Augustine's initial Manichean influences and see how these beliefs carry over to his mature thought, which is evident in the seventh book of the Confessions. After Augustine's time with the Manicheans, we look at how he was so influenced by the books of the Platonists (libri platonicorum). Although Augustine's position regarding the problem of evil shifts, his idea of the primacy of (...) the soul is still evident in his thought process. To wit, Augustine posits that evil must be considered a privation of the Good, so much so as to reach the point of complete nonentity. Human beings' ability to be corrupted by evils rests in their position as being created ex-nihilo by God. With this creation also comes an inherent mutability. Due to human mutability, Augustine believes that God is not responsible for such evil actions. This paper also contrasts this belief with modern empiricist David Hume's idea regarding God's responsibility for human actions. Hume argues that the volition of all human actions rests in God as Creator of the world. As creator, Hume claims that God places human beings in a position to act. If humans are predisposed to perform evil actions, they cannot be faulted. Augustine would counter that argument by claiming that evil is not a substance. Not being a substance, evil is there not ascribable to God. Ultimately, Augustine's theodicy is based upon the goodness of God. (shrink)

Kevin Gray,Kafumu Kalyalya | : In this paper we challenge the role of consent in the global order by discussing current modes of international law making in the global order. We contend that the features of state consent in international law depart substantially from those assumed by theorists of the liberal order, who subscribe, in most cases, to the realist conception of state action. We argue, against those theorists, that state consents to coercive measures, and the state’s role (...) in carrying them out, has ceased to be central to an account of global law. We conclude that international law—often thought of as law beyond the state—now has expanded its scope to reach individuals and corporations, and that this change has important ramifications for theories of global justice. | : Dans cet article, nous interrogeons le rôle de l’accord dans l’ordre global en discutant les modes courants de constitution des lois internationales au sein de cet ordre. Nous prétendons que les fonctions de l’accord étatique en loi internationale s’écartent substantiellement de celles assumées par les théoriciens de l’ordre libéral, qui souscrivent, dans la plupart des cas, à la conception réaliste de l’action étatique. Contre ces théoriciens, nos posons que l’État consent à des mesures coercitives, et que le rôle de l’État d’effectuer celles-ci a cessé d’être central dans l’explication de la loi globale. Nous concluons que la loi internationale – souvent pensée comme une loi hors de l’État – a maintenant étendu son spectre jusqu’aux individus et corporations, et que ce changement a d’importantes ramifications que les théories globales de la justice devraient considérer. (shrink)

In this paper, I consider the difficult relationship between Rawls, religion and the values that religious believers might consider important in order to lead the good life. Contrary to many of Rawls’ defenders, I argue that at least some of the values that religious citizens are likely to hold cannot be accounted for under Rawls’ theory or under his conception of the good life. I argue that the model of goods which Rawls takes to be part of a thin theory (...) of the good is tied to his belief that under the Original Position justice can be derived from calculations of self-interest alone. To perform my critique, I consider the paradigmatic case of honour in so-called traditional societies. I argue that the way Rawls thematizes primary goods in A Theory of Justice , including concepts like esteem, cannot account for the way honour manifests itself inside traditional communities. I conclude the paper by considering how Rawls might be able to defend his theory against my objection, by considering the relationship between Rawls’ theory, and the rationalization and secularization of society. (shrink)

This paper argues that cosmopolitan law has been more successfully achieved not by appeal to a supra-state authority or community, but by the development of features of existing treaty law. Specifically, it shows how the International Criminal Court's jurisdiction over serious human rights violations has been extended to the citizens and territories of non-member states – and even to otherwise immune state officials – not by challenging the sovereignty of non-member states directly, but on the basis of member states’ own (...) territorial sovereignty and the universal jurisdiction which they delegate to the Court and to the United Nations Security Council. In light of this, the authors argue that cosmopolitanism is better conceived not as invoking an independent sense of global community that supersedes and constrains state sovereignty, but as an immanent, contingent and creative development of statist criminal law itself, rooted in its principles of state sovereignty. (shrink)

Taking Habermas’s Die nachholende Revolution as a foil, I contend that in his discussions of 1989, Habermas has misunderstood the nature of the anti-Communist revolutions. Comparing them to his writings on the public sphere and the student protest movements in Germany, I argue that the revolutions do not represent the triumph of capitalism anymore than they represent the triumph of Western democracy. Calling the events catch-up revolutions is to frame the events as the expansion of modernity and nothing more. Rather, (...) the revolutions show that the revolutionaries in Eastern Europe were grappling with the same problem, namely the control of technical subsystems, that the students and revolutionaries of 1968 were. Viewed in the light of Habermas’s writings from the 1960s, we end up better understanding the twin extremes of capitalism and bureaucratic-totalitarianism. (shrink)

Taking Habermas’s Die nachholende Revolution as a foil, I contend that in his discussions of 1989, Habermas has misunderstood the nature of the anti-Communist revolutions. Comparing them to his writings on the public sphere and the student protest movements in Germany, I argue that the revolutions do not represent the triumph of capitalism anymore than they represent the triumph of Western democracy. Calling the events catch-up revolutions is to frame the events as the expansion of modernity and nothing more. Rather, (...) the revolutions show that the revolutionaries in Eastern Europe were grappling with the same problem, namely the control of technical subsystems, that the students and revolutionaries of 1968 were. Viewed in the light of Habermas’s writings from the 1960s, we end up better understanding the twin extremes of capitalism and bureaucratic-totalitarianism. (shrink)

In this paper I argue that the attempt to develop practice dependence as a middle path between cosmopolitanism and social liberalism ultimately fails. In particular, I contend that the only type of practice dependence that can coherently be defended collapses into cosmopolitanism. I trace the origin of practice dependence theory from Cohen and Sabel’s work through Sangiovanni and Valentini. After discussing the various different types of practice dependence, I argue that the only type of practice dependence that stands up to (...) scrutiny is functionalism, but that if functionalism is true, it commits us to cosmopolitanism. (shrink)

Although the focus of "Globalizing Democracy and Human Rights" is practical, Gould does not shy away from hard theoretical questions, such as the relentless debate over cultural relativism, and the relationship between terrorism and democracy.