The evolution and maintenance of cooperation in human and animal societies challenge various disciplines ranging from evolutionary biology to anthropology, social sciences, and economics. In social interactions, cooperators increase the welfare of the group at some cost to themselves whereas defectors attempt to free ride and neither provide benefits nor incur costs. The problem of cooperation becomes even more pronounced when increasing the number of interacting individuals. Punishment and voluntary participation have been identified as possible factors to support cooperation and (...) prevent cheating. Typically, punishment behavior is unable to gain a foothold in a population, while volunteering alone can efficiently prevent deadlocks in states of mutual defection but is unable to stabilize cooperation. The combined effects of the two mechanisms have surprisingly different consequences in finite and infinite populations. Here we provide a detailed comparison of the two scenarios and demonstrate that driven by the inherent stochasticity of finite populations, the possibility to abstain from social interactions plays a pivotal role, which paves the way for the establishment of cooperation and punishment. (shrink)

We continue the exploration of various aspects of divisibility of ultrafilters, adding one more relation to the picture: multiplicative finite embeddability. We show that it lies between divisibility relations \ and \. The set of its minimal elements proves to be very rich, and the \-hierarchy is used to get a better intuition of this richness. We find the place of the set of \-maximal ultrafilters among some known families of ultrafilters. Finally, we introduce new notions of largeness of (...) subsets of \, and compare it to other such notions, important for infinite combinatorics and topological dynamics. (shrink)

Aggregative consequentialism and several other popular moral theories are threatened with paralysis: when coupled with some plausible assumptions, they seem to imply that it is always ethically indifferent what you do. Modern cosmology teaches that the world might well contain an infinite number of happy and sad people and other candidate value-bearing locations. Aggregative ethics implies that such a world contains an infinite amount of positive value and an infinite amount of negative value. You can affect only (...) a finite amount of good or bad. In standard cardinal arithmetic, an infinite quantity is unchanged by the addition or subtraction of any finite quantity. So it appears you cannot change the value of the world. Modifications of aggregationism aimed at resolving the paralysis are only partially effective and cause severe side effects, including problems of “fanaticism”, “distortion”, and erosion of the intuitions that originally motivated the theory. Is the infinitarian challenge fatal? (shrink)

The problem of how to rationally aggregate probability measures occurs in particular when a group of agents, each holding probabilistic beliefs, needs to rationalise a collective decision on the basis of a single ‘aggregate belief system’ and when an individual whose belief system is compatible with several probability measures wishes to evaluate her options on the basis of a single aggregate prior via classical expected utility theory. We investigate this problem by first recalling some negative results from preference and judgment (...) aggregation theory which show that the aggregate of several probability measures should not be conceived as the probability measure induced by the aggregate of the corresponding expected utility preferences. We describe how McConway’s :410–414, 1981) theory of probabilistic opinion pooling can be generalised to cover the case of the aggregation of infinite profiles of finitely additive probability measures, too; we prove the existence of aggregation functionals satisfying responsiveness axioms à la McConway plus additional desiderata even for infinite electorates. On the basis of the theory of propositional-attitude aggregation, we argue that this is the most natural aggregation theory for probability measures. Our aggregation functionals for the case of infinite electorates are neither oligarchic nor integral-based and satisfy a weak anonymity condition. The delicate set-theoretic status of integral-based aggregation functionals for infinite electorates is discussed. (shrink)

According to one of Leibniz's theories of contingency a proposition is contingent if and only if it cannot be proved in a finite number of steps. It has been argued that this faces the Problem of Lucky Proof , namely that we could begin by analysing the concept ‘Peter’ by saying that ‘Peter is a denier of Christ and …’, thereby having proved the proposition ‘Peter denies Christ’ in a finite number of steps. It also faces a more (...) general but related problem that we dub the Problem of Guaranteed Proof . We argue that Leibniz has an answer to these problems since for him one has not proved that ‘Peter denies Christ’ unless one has also proved that ‘Peter’ is a consistent concept, an impossible task since it requires the full decomposition of the infinite concept ‘Peter’. We defend this view from objections found in the literature and maintain that for Leibniz all truths about created individual beings are contingent. (shrink)

We say that an n-argument predicate P n is finite, if P is a finite set. Note that the set of individuals is infinite! Finite predicates are useful in data bases and in finite mathematics. The logic DBL proposed here operates on finite predicates only. We construct an imbedding for DBL in a special modal logic MPL. We prove that if a finite predicate is expressible in the classical logic, it is also expressible (...) in DBL. Quantifiers are not necessary in DBL. Some simple algebraic properties of DBL are indicated. (shrink)

Although Alain Badiou dedicates a number of texts to the philosophy of Benedict de Spinoza throughout his work—after all, the author of a systematic philosophy of being more geometrico must be a point of reference for the philosopher who claims that “mathematics = ontology”—the reading offered in Meditation Ten of his key work Being and Event presents the most significant moment of this engagement. Here, Badiou proposes a reading of Spinoza’s ontology that foregrounds a concept that is as central to, (...) and celebrated in, his philosophy as it is strictly excluded by Spinoza: the void. In nuce, Badiou contends that for all of Spinoza’s efforts to offer an ontology of total plenitude, the void returns in his philosophy under the (at first sight) unlikely name of infinite mode. The presence of this errant name in Spinoza’s philosophy bears witness to the failure of his most profound intellectual endeavour. However striking Badiou’s reading of Spinoza, this paper argues that it fails to adequately grasp Spinoza’s metaphysics, particularly with respect to the central concept of modal essence, a concept which does not appear at all in the Badiouian text. By introducing a consideration of this concept, it becomes able to resolve the status of infinite modes, and to account for the move across the notorious finite–infinite divide. Thus the argument turns to the reading of Spinoza offered by Gilles Deleuze for a more thorough-going and nuanced approach, much superior to Badiou’s procrustean critique. (shrink)

The question of whether it is possible to philosophize outside the categories of rationalist philosophy is not limited to methodology. It has ideological overtones. Namely, the rationalism that has developed in philosophy in modern times, after Descartes, is inevitably supplemented by mechanics. The world is seen as a machine, the living is reduced to mechanisms. Rationalism becomes a machine of mentality. Taking it as a model of normal thinking, giving it a universal value, we thereby impose Western way of thinking (...) on other cultures with a different mentality. The question, therefore, is not about the method of scientific knowledge but about the power realized through the transformation of mentality. Scientific knowledge conquers space. Philosophy produces a transformation of mentality. New rationalism, according to G. Bachelard, is in dialectical relation with the usual realism of the natural Sciences. In this process, there is the implementation of new schemes developed in the complex interaction of “ratio” and experiment. Modern science is the collective creation of a new reality. This is the reality of the “picture of the world.” There is another view of science, rationalism and modern philosophy. According to him, rationalism is dualism, atomism and social individualism. These features reveal the anti-traditional and anti-metaphysical spirit of modern Western civilization. The traditional approach is non-duality, unity provided by metaphysical principles. The new rationalism in science is characterized by the belief in the infinite progress of knowledge. But knowledge itself, from the point of view of tradition, is devoid of great value. Philosophy uses categories of ancient Greek philosophy or German classical philosophy, but they cannot be considered universal. They corresponded not only to the time and place, the conditions in which they arose, but also to the characteristic mentality of the peoples and individuals who created them. They are not suitable for understanding the metaphysical doctrines of traditional cultures. The language of traditional cultures is symbolic and synthetic in contrast to the analytical language of Western science and philosophy. (shrink)

Familiar proof theoretical and especially automated deduction methods sometimes accept infinity where, in fact, it can be omitted. Our first example deals with the infinite supply of individual variables admitted in 1-order deductions, the second one deals with infinite-branching rules in sequent calculi with number-theoretical induction. The contents of Section 1 summarize and extend basic ideas and results published elsewhere, whereas basic ideas and results of Section 2 are exposed for the first time in the present paper. We (...) consider classical formalisms only, to which constructive formalisms can be reduced by Shanin-style interpretations. (shrink)

The epochal transition from the medieval to the modern world has long been thought in terms of the "infinitization" of the world-picture, that is, as the transition from the finite, hierarchically ordered medieval cosmos to the infinite and homogeneous universe of the new astronomy and physics. In this dissertation I argue that this process of "infinitization" must be understood intensively as well as extensively. Nature, in the modern age, is thought not only as infinitely extended in space, but (...) also as exhibiting an infinite richness in all of its parts. Each individual being is grasped as utterly unique and consequently as conceptually inexponable. Thus the infinitization of the real leads to an infinitization of the knowable-- the radical shift in ontology grounds a corresponding shift in epistemology, so that the progress of human knowledge is understood as an unending project infinitely extended over time. ;I take as my point of departure a consideration of Hans Blumenberg's reading of this epochal transition in The Legitimacy of the Modern Age. Blumenberg views the transference of the attribute of infinity from God to the world as characteristic of the modern rejection of the medieval tendency to locate the measure of self- and world-understanding in a transcendent deity. The "worldliness" of the modern age, Blumenberg holds, must be understood as a reaction to the extreme intensification of divine transcendence in late medieval Nominalism. In the face of an absconded God, human "self-assertion" aims at the realization of this-worldly possibilities through the mastery and alteration of reality. I agree with Blumenberg that the primary expression of this self-assertion has been seen in the progress of modern science. The possibility of that progress, however, is grounded in an understanding of nature as a law-like and yet inexhaustible field of investigation directing thought toward an objectivity that is never entirely realized. The origin of such an understanding, I argue, is to be found in late medieval Neoplatonic speculation on the nature of God's omnipresence in the world, in particular in the notion of the immanence of the infinite in the finite. (shrink)

We present an abstract social aggregation theorem. Society, and each individual, has a preorder that may be interpreted as expressing values or beliefs. The preorders are allowed to violate both completeness and continuity, and the population is allowed to be infinite. The preorders are only assumed to be represented by functions with values in partially ordered vector spaces, and whose product has convex range. This includes all preorders that satisfy strong independence. Any Pareto indifferent social preorder is then shown (...) to be represented by a linear transformation of the representations of the individual preorders. Further Pareto conditions on the social preorder correspond to positivity conditions on the transformation. When all the Pareto conditions hold and the population is finite, the social preorder is represented by a sum of individual preorder representations. We provide two applications. The first yields an extremely general version of Harsanyi's social aggregation theorem. The second generalizes a classic result about linear opinion pooling. (shrink)

Transfer of information between senders and receivers, of one kind or another, is essential to all life. David Lewis introduced a game theoretic model of the simplest case, where one sender and one receiver have pure common interest. How hard or easy is it for evolution to achieve information transfer in Lewis signaling?. The answers involve surprising subtleties. We discuss some if these in terms of evolutionary dynamics in both finite and infinite populations, with and without mutation.

In lieu of an abstract, here is a brief excerpt of the content:Bishop Berkeley Exorcises the Infinite: Fuzzy Consequences of Strict Finitism1 David M. Levy Introduction It all began simply enough when Molyneux asked the wonderful question whether a person born blind, now able to see, would recognize by sight what he knew by touch (Davis 1960). After George Berkeley elaborated an answer, that we learn to perceive by heuristics, the foundations ofcontemporarymathematics wereinruin. Contemporary mathematicians waved their hands and (...) changed the subject.2 Berkeley's answer received a much more positive response from economists. Adam Smith,in particular, seizedupon Berkeley's doctrine that welearn to perceive distance tobuild anelaborate system in which one learns to perceive one's self-interest.3 Perhaps because older histories of mathematics are a positive hindrance in helping us understand the importance of Berkeley's argument against infinitesimals,4 its consequences for economics have passed unnoticed. If infinitesimal numbers are ruled out, and we wish to maintain an algebra, then we rule out infinite numbers. This implication of Berkeley's argumenthas a dramatic implication for our understanding ofthe history ofideas about rule-directed action. Let me motivate this claim with a well-known historical problem. The social doctrine set forth by John Locke outside the Two Treatises depends critically upon individual belief about possible states of infinite pain or pleasure. The critical issue for the intolerance of atheism in Locke's system is his claim that without beUefin the infinite bliss of heaven forgone by a criminal, there is no reason to think a calculating individual will avoid crime.5 Locke's discussion is so different from the discussion of religion by David Hume and Smith. Here, questions of the substance of belief—the infinite worth of Heaven—have largely vanished.6 What happened between Locke and Hume-Smith? One answer is Pierre Bayle and Bernard Mandeville. Berkeley's doctrine that only the finite is perceived is, I think, a better answer. Out of sight, out ofmind. In this paper I propose to give a close look at the content and consequences ofBishop Berkeley's once famous Towards a New Theory ofVision. While there are other aspects ofBerkeley's work that attract Volume XVIII Number 2 511 DAVID M. LEVY attention from historians ofeconomics,7 1 shall claim that Berkeley's insight into human perception in the Theory of Vision is central to Adam Smith's attempt to found economic behaviour on the self-awareness of systematic illusion. There is no doubt that Smith finds that much ofhuman behaviour is characterized by illusion.8 The difficulty arises, or so it seems to me, from modern unwilUngness to beheve that illusion can be systematic. There are four pieces to my argument. First, we consider Berkeley's statementofwhat we shall call his doctrine ofstrict finitism in perception.9 Berkeley puts forward two postulates: 1) there exists some minimum perceptible quantity, and 2) distance is not perceived directly but by experience. Second, we define strict finitism and demonstrate that it translates into the claim that perception is fuzzy in a technical sense. If Berkeleian perception is fuzzy, then some modern criticism of Berkeley's strict finitism is based upon a simple misunderstandingofthe propertiesoffuzzyrelations. Thisis quite easy to set right. (Getting right the doctrine of those who implicitly take perception to be fuzzy—Adam Smith is my prime candidate—will be muchharder.)Third, wereconsiderBerkeley's dispute withMandeville on the role ofinfinite gain in choice. Here, we find Mandeville putting forward the Berkeleian position that if infinite amounts mattered, people would not behave as they do. Berkeley's position against Mandeville, however, seems to depend upon the supposition that infinite distance is sensible.10 Earlier, however, the position that Mandeville argued for was defended by Berkeley. Fourth, we ask why all this isn't well known. The debate between Berkeley and Mandeville has been well studied, as has been the link between Berkeley and the Scots. In fact, our work focuses on one aspect of Berkeley's work that led to his challenge of the contemporary foundations of the calculus. If sense is a matter of perception, and infinitesimals cannot be perceived, then they are quite literally nonsense. Armed with this insight, Berkeley refuted contemporary mathematics. Nevertheless, the calculus was too... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Bishop Berkeley Exorcises the Infinite: Fuzzy Consequences of Strict Finitism1 David M. Levy Introduction It all began simply enough when Molyneux asked the wonderful question whether a person born blind, now able to see, would recognize by sight what he knew by touch (Davis 1960). After George Berkeley elaborated an answer, that we learn to perceive by heuristics, the foundations ofcontemporarymathematics wereinruin. Contemporary mathematicians waved their hands and (...) changed the subject.2 Berkeley's answer received a much more positive response from economists. Adam Smith,in particular, seizedupon Berkeley's doctrine that welearn to perceive distance tobuild anelaborate system in which one learns to perceive one's self-interest.3 Perhaps because older histories of mathematics are a positive hindrance in helping us understand the importance of Berkeley's argument against infinitesimals,4 its consequences for economics have passed unnoticed. If infinitesimal numbers are ruled out, and we wish to maintain an algebra, then we rule out infinite numbers. This implication of Berkeley's argumenthas a dramatic implication for our understanding ofthe history ofideas about rule-directed action. Let me motivate this claim with a well-known historical problem. The social doctrine set forth by John Locke outside the Two Treatises depends critically upon individual belief about possible states of infinite pain or pleasure. The critical issue for the intolerance of atheism in Locke's system is his claim that without beUefin the infinite bliss of heaven forgone by a criminal, there is no reason to think a calculating individual will avoid crime.5 Locke's discussion is so different from the discussion of religion by David Hume and Smith. Here, questions of the substance of belief—the infinite worth of Heaven—have largely vanished.6 What happened between Locke and Hume-Smith? One answer is Pierre Bayle and Bernard Mandeville. Berkeley's doctrine that only the finite is perceived is, I think, a better answer. Out of sight, out ofmind. In this paper I propose to give a close look at the content and consequences ofBishop Berkeley's once famous Towards a New Theory ofVision. While there are other aspects ofBerkeley's work that attract Volume XVIII Number 2 511 DAVID M. LEVY attention from historians ofeconomics,7 1 shall claim that Berkeley's insight into human perception in the Theory of Vision is central to Adam Smith's attempt to found economic behaviour on the self-awareness of systematic illusion. There is no doubt that Smith finds that much ofhuman behaviour is characterized by illusion.8 The difficulty arises, or so it seems to me, from modern unwilUngness to beheve that illusion can be systematic. There are four pieces to my argument. First, we consider Berkeley's statementofwhat we shall call his doctrine ofstrict finitism in perception.9 Berkeley puts forward two postulates: 1) there exists some minimum perceptible quantity, and 2) distance is not perceived directly but by experience. Second, we define strict finitism and demonstrate that it translates into the claim that perception is fuzzy in a technical sense. If Berkeleian perception is fuzzy, then some modern criticism of Berkeley's strict finitism is based upon a simple misunderstandingofthe propertiesoffuzzyrelations. Thisis quite easy to set right. (Getting right the doctrine of those who implicitly take perception to be fuzzy—Adam Smith is my prime candidate—will be muchharder.)Third, wereconsiderBerkeley's dispute withMandeville on the role ofinfinite gain in choice. Here, we find Mandeville putting forward the Berkeleian position that if infinite amounts mattered, people would not behave as they do. Berkeley's position against Mandeville, however, seems to depend upon the supposition that infinite distance is sensible.10 Earlier, however, the position that Mandeville argued for was defended by Berkeley. Fourth, we ask why all this isn't well known. The debate between Berkeley and Mandeville has been well studied, as has been the link between Berkeley and the Scots. In fact, our work focuses on one aspect of Berkeley's work that led to his challenge of the contemporary foundations of the calculus. If sense is a matter of perception, and infinitesimals cannot be perceived, then they are quite literally nonsense. Armed with this insight, Berkeley refuted contemporary mathematics. Nevertheless, the calculus was too... (shrink)

Renowned scholar Robert Adams explores the relation between religion and ethics through a comprehensive philosophical account of a theistically-based framework for ethics. Adams' framework begins with the good rather than the right, and with excellence rather than usefulness. He argues that loving the excellent, of which adoring God is a clear example, is the most fundamental aspect of a life well lived. Developing his original and detailed theory, Adams contends that devotion, the sacred, grace, martyrdom, worship, vocation, faith, and other (...) concepts drawn from religious ethics have been sorely overlooked in moral philosophy and can enrich the texture of ethical thought. (shrink)

A central difficulty in the way of understanding Spinoza's metaphysical system is that of reconciling two apparently contradictory theories of the causation of finite modes found in his Ethics. The easiest way to present the problem is to place these two accounts side by side.A. All things which follow from the absolute nature of any attribute of God must forever exist, and must be infinite; that is to say, through that attribute they are eternal and infinite. A (...) thing which has been determined to any action was necessarily so determined by God. Together, these propositions entail that God is a necessary and sufficient condition for the occurrence of any action, and that anything so determined is eternal and infinite. Compare these implications with the following.B. An individual thing or a thing which is finite and has determinate existence cannot exist nor be determined to action unless it be determined to existence and action by another cause which is also finite and has a determinate existence; and again this cause cannot exist nor be determined to action unless by another cause which is also finite and determined to existence and action, and so on, ad infinitum. (shrink)

The aim of this study is to try to make use of real numbers for representing an infinite analysis of individual notions in an infinity of possible worlds.As an introduction to the subject, the author shows, firstly, the possibility of representing Boole's lattice of universal notions by an associate Boole's lattice of rational numbers.But, in opposition to the universal notions, definable by a finite number of predicates, an individual notion, cannot admits this sort of definition, because each state (...) of an individual subject is characterized by the values taken by an infinite number of predicates, each of whom may appear or disappear in the next state.The notion of “degree of identification of an individual notion” is then introduced and arithmetized by a rational number.As an individual notion can be defined by a convergent succession of degrees of identification, the real characteristic number of such an individual notion can be defined by the corresponding convergent succession of rational numbers, satisfying Cauchy's conditions for the convergence of successions. (shrink)

This book presents a new theory of the nature of the space in which substantial, enduring objects (objects that are identifiable for more than a fleeting instant) connect with each other and cohere within themselves. This posited fundamental space underlies the common perception of space as necessarily having to identify its contents by separating them within finite beginning and ending boundaries. In the real space each entity is positive and is directly connected to every entity. These connections differ depending (...) on the spatial and temporal dimensions in which they are defined and viewed. This offers a radically new solution to the mind-body problem, which rechearchers in mind-brain theory have not yet satisfactorily solved. And it resolves the conflicting interpretations of quantum physics vs. classical relativistic physics. Even more encouraging to readers who are not specialists in the sciences are the implications for realizing enduring values and a greater appreciation of excellence in our work, education, avocations, and the kindof entertainment we seek. (shrink)

Introduction. The article studies some aspects of the phenomenon of human individuality. Human individuality is a subject of research interest in many sciences. In the process of human development, the inner world of human individuality, unique in its multidimensionality of consequences, began to be exposed, it is a form of human subjective being, and exists and acts autonomously, while maintaining its integrity. The purpose of the study is to identify basic approaches to defining the ontological essence of a person’s individuality. (...) Research methods: historical and analytical method, conceptual and terminological analysis, analysis, synthesis, generalization, abstraction and concretization, comparison, etc. Scientific novelty of the study. The article provides a definition of the phenomenon of «human individuality» from the standpoint of different interdisciplinary directions. The authors note that individuality is not a self-contained integrity that arbitrarily constructs an external and internal manifestation, that is the subjective world of human reality, but rather resembles a space in which the finite and infinite collide, and the opacity of semantic content collide with the content which has already been manifested. Research results: The paper presents various approaches to the phenomenon of individuality from the standpoint of different worldview positions. The authors note that individuality is not a selfcontained integrity that arbitrarily constructs its external and internal manifestation - the subjective world of human reality. Conclusions. The authors note that the essence of individuality is not an alienated sign of the nature of a specific event in the history of evolution. Individuality provides a person with a high opportunity to become a thought of being. In this situation, a person, of course, feels himself an ontological unit responsible for being, realizing himself as a bearer and participant in the meanings of being In the process of understanding his individuality, a person is aware of himself as a subject of the historical process, civil history, potentially possessing all the characteristics of an individualized human race, the ways he unfolds a multidimensional individuality in a generational space - the time of eras, processes that have distinguished a person from living nature. (shrink)

Adams offers a theistically-based framework for ethics, based upon the idea of a transcendent, infinite good, which is God, and its relation to the many finite examples of good in our experience. His account shows how philosophically unfashionable religious concepts can enrich ethical thought. "...one of the two most important books in moral philosophy of the last quarter century, the other being After Virtue."--Theology Today.

This is the preface for a special volume published by the Journal of Applied Non-Classical Logics Volume 9, Issue 1, 1999.

In Finite and Infinite Goods, Adams develops a sophisticated and richly detailed Platonic-theistic framework for ethics. The view is Platonic in virtue of being Good-centered; it is theistic both in identifying God with the Good and, more distinctively, in including a divine command theory of moral obligation. Readers familiar with Adams’s earlier divine command theory will recall that in response to the worry that God might command something evil, Adams introduced an independent value constraint, claiming that only the (...) commands of a loving God were fit to constitute moral obligations. In Finite and Infinite Goods he develops this notion of a loving and good God in what is now a fundamentally Good-centered ethical framework with a subordinate divine command theory of moral obligation. There is much of worth here, especially for theists wishing to think through the merits of a good-based theistic ethics, but also for those nontheists like myself who have a general interest in the nature of value and obligation. (shrink)

In this paper we deepen Mundici's analysis on reducibility of the decision problem from infinite-valued ukasiewicz logic to a suitable m-valued ukasiewicz logic m , where m only depends on the length of the formulas to be proved. Using geometrical arguments we find a better upper bound for the least integer m such that a formula is valid in if and only if it is also valid in m. We also reduce the notion of logical consequence in to the (...) same notion in a suitable finite set of finite-valued ukasiewicz logics. Finally, we define an analytic and internal sequent calculus for infinite-valued ukasiewicz logic. (shrink)

This paper traces Schelling’s discussions of individuation from the 1799 Erster Entwurf eines Systems der Naturphilosophie to the 1802 dialogue, Bruno. It argues that the Erster Entwurf is unable to solve what Schelling there calls “the highest problem of the philosophy of nature,” because nature as pure productivity necessarily tends to annihilate all individuality. It is only in 1801 and 1802, the years that mark Schelling’s construction of an Identitätssystem, that a solution emerges. This solution is based on the (...) rejection of one of the defining orthodoxies of German Idealism, Spinoza’s dictum that omnis determinatio est negatio, and on the subsequent theorization of a form of infinite finitude. Consideration of Schelling’s changing attitude to the problem of individuation is intended as a case study of the more general shift from Naturphilosophie to Identitätssystem. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Transitions to a Modern Cosmology: Meister Eckhart and Nicholas of Cusa on the Intensive InfiniteElizabeth BrientThe Epochal Transition from the late medieval to the early modern world has long been thought in terms of the gradual “infinitization” of the cosmos. Traditionally this process has been studied by focusing on the pre-history and the aftermath of the Copernican revolution, that is, by describing the transition from the finite, hierarchically (...) ordered medieval cosmos to the infinite and homogeneous universe of the new astronomy. I would like to situate this paper within the context of a more general reading of the process of the infinitization of the medieval cosmos. That is, I want to argue that this process of “infinitization” must be understood intensively as well as extensively. Nature, in the modern age, is thought of not only as infinitely extended in space, but also as exhibiting an infinite richness in all of its parts. Each particular, each individual being is grasped as utterly unique, as infinitely rich and consequently as conceptually inexponable. Thus the infinitization of the real leads to an infinitization of the knowable—the radical shift in ontology grounds a corresponding shift in epistemology, so that the progress of human knowledge is understood as an unending project infinitely extended over time. In order to underscore both aspects of the infinitization of the medieval cosmos (intensive as well as extensive) I will focus on the way in which a single metaphor (that of the infinite sphere) is unfolded historically in the thought of Meister Eckhart and Nicholas of Cusa.1 This paper may thus be viewed as a case study of [End Page 575] the way in which the modern notion of “intensive infinity” may be traced back to late medieval speculation on the nature of God’s omnipresence in the world.1. EARLY SOURCES OF THE INFINITE SPHERE METAPHORThe metaphor of the infinite sphere makes its first appearance in the pseudo-Hermetic Liber XXIV philosophorum, a small book in the literary form of an “opus propositionum” which, in twenty-four propositions, describes the ineffable essence of God in the mystical language of Neoplatonic theology. It is in the second proposition of this book that we find for the first time the formulation, “God is an infinite sphere, whose center is everywhere, whose circumference is nowhere.”2 This formulation represents a significant development in the history of the Neoplatonic conception of intensive unity, and in the Christian understanding of divine infinity. The Neoplatonic One is here identified with God as infinite being. As a result the relationship between the One and the many, between God and the world, comes to be viewed in a new light, through the lens of “infinite unity.” The metaphor of the infinite sphere is a paradoxical formulation which pictures the coincidence of divine immanence with divine transcendence. That is, it expresses a double infinity: both intensive and extensive. God’s center is located at every point, as it were, indicating the plenitude and infinite richness of God’s being, while there is no limit, no boundary to the extent of God’s all-encompassing reach. The infinite sphere figures the divine nature as an intensive plenitude without end or bounds. The third and next proposition in this same work goes on to strengthen this sense of divine intensity, by asserting that God in his entirety is present in every part (Deus est totus in quolibet sui). The divine nature is not spread out compositionally, as it were, but is wholly present throughout. Proposition 18 describes God as a sphere which has just as many circumferences as points (Deus est sphaera cuius tot sunt circumferentiae, quot sunt puncta), once again, an exploding image which can only make sense in the case of an infinite sphere. These three propositions taken together provide us with a springboard for considering the nature of infinite unity. The exploding paradox of the infinite sphere [End Page 576] provides a rubric for thinking the coincidence of transcendence and immanence, a way of conceptualizing infinite unity, which cannot be “grasped” with finite concepts or through discursive reasoning. The metaphor of the infinite sphere finds its meaning precisely in the way... (shrink)

We agree with Hilbert’s assessment that the concept of ‘infinite’ stands in need of clarification – however, our proposed ‘solution’ is almost diammetrically opposed to that of Hilbert.

By operations on models we show how to relate completeness with respect to permissivenominal models to completeness with respect to nominal models with finite support. Models with finite support are a special case of permissive-nominal models, so the construction hinges on generating from an instance of the latter, some instance of the former in which sufficiently many inequalities are preserved between elements. We do this using an infinite generalisation of nominal atoms-abstraction. The results are of interest in (...) their own right, but also, we factor the mathematics so as to maximise the chances that it could be used off-the-shelf for other nominal reasoning systems too. Models with infinite support can be easier to work with, so it is useful to have a semi-automatic theorem to transfer results from classes of infinitely-supported nominal models to the more restricted class of models with finite support. In conclusion, we consider different permissive-nominal syntaxes and nominal models and discuss how they relate to the results proved here. (shrink)

We describe the progress of model theory in the last half century from the standpoint of how finite model theory might develop.

When I undertook to write an article for mathematical laymen on the mathematical infinite. I did not realize the depths of my own layness, I do now. Having refreshed my memory of the classics of infinity by re-reading among other things the famous papers of Cantor and Zermelo, and having struggled like a boa constrictor to swallow the latest papal bull on the human significance of the infinite, I am completely reduced to what Professor E. W. Hobson aptly (...) and somewhat indignantly calls Mathematical Nihilism. (shrink)

This is a review of Robert Merrihew Adams's book Finite and Infinite Goods (Oxford UP 1999), which provides an impressive theistic axiological and ethical theory.

The paper is concerned with the problem of individuation in Spinoza. Spinoza's account of individuation leads to the apparent contradiction between, on the one hand, the view that substance (God or Nature) is simple, eternal, and infinite, and on the other, the claim that substance contains infinite differentiation - determinate and finite modes, i.e. individuals. A reconstruction of Spinoza's argument is offered which accepts the reality of the contradiction and sees it as a consequence of (...) Spinoza's way of posing the problem of individuation: it is argued that Spinoza's ontology is constructed on the basis of his methodology, rather than conversely. The contradiction appears in its acutest form in Spinoza's discussion of number. The paper explains the philosophical motivation behind Spinoza's Problematik and reflects on the historical context of the contradiction to which that Problematik gives rise. (shrink)

continent. 1.2 (2011): 78-91. This article consists of three parts. First, I will review the major themes of Quentin Meillassoux’s After Finitude . Since some of my readers will have read this book and others not, I will try to strike a balance between clear summary and fresh critique. Second, I discuss an unpublished book by Meillassoux unfamiliar to all readers of this article, except those scant few that may have gone digging in the microfilm archives of the École normale (...) supérieure. The book in question is Meillassoux’s revised doctoral dissertation L’Inexistence divine (or The Divine Inexistence ), with its seemingly bizarre vision of a God who does not yet exist but might exist in the future. Without literally accepting this view, I will claim that it is philosophically interesting in ways that even a hardened sceptic might be able to appreciate. Third and finally, I will speculate on the possible future of Meillassoux’s speculative materialism itself. And here I mean its future development not by Meillassoux, but by those readers who might be inspired by his book. Plato could never have predicted the emergence of Aristotle’s philosophy, despite the obvious debt of the latter to the former. Nor could Descartes have predicted Spinoza and Leibniz, nor Kant the German Idealists, and neither could Husserl in 1901 have foreseen the later emergence of Heidegger. How are the works of interesting philosophers transformed by later thinkers of comparable importance? While it may seem that there are countless ways to do this, I think there are only two basic ways in which this happens: you can radicalize your predecessors, or you can reverse them. I will close this article with a few words about these two methods, and try to imagine how Meillassoux might be radicalized or reversed by some future admirer. My view is that the more important thinkers are, the easier they are to radicalize or reverse. This helps explain why the great philosophers of the West have so often appeared in clusters, succeeding one another at relatively brief intervals during periods of especial ferment. 1. After Finitude After Finitude is unusually short for such an influential book of philosophy: running to just 178 pages in the original French, and an even more compact 128 pages in the English version, despite the introduction of roughly eight pages of new material for the English edition. Rather than summarizing Meillassoux’s book in the order he intended, I will focus on six points that strike me as the pillars of his debut book. Along the way, I will offer a few criticisms as well. The first pillar of the book is Meillassoux’s own term “correlationism.”1 Although he introduces this term as the name for an enemy, it is striking that Meillassoux remains impressed by correlationism much more than his fellow speculative realists are. This continued appreciation for his great enemy influences the shape of his own ontology. Is there a world outside our thinking of it, or does the world consist entirely in being thought? Traditionally, this dispute between realism and idealism has been dismissed in continental philosophy as a “pseudo-problem,” in a strategy pioneered by Husserl and extended by Heidegger. We cannot be realists, since following Kant we have no direct access to things-in-themselves. But neither are we idealists, since the human being is always already outside itself, aiming at objects in intentional experience, deeply engaged with practical implements, or stationed in some particular world-disclosing mood. The centuries-old dispute between realism and idealism is dissolved by saying that we cannot think either real or ideal in isolation from the other. There is neither human without world nor world without human, but only a primordial correlation or rapport between the two. This is what “correlationism” means: philosophy trapped in a permanent meditation on the human-world correlate, trying to find the best model of the correlate: is it language, intentionality, embodiment, or some other form of correlation between human and world? Among other problems, this generates some friction between philosophy and the literal meaning of science. When cosmologists say that the universe originated 13.5 billion years ago, they do not mean “13.5 billion years ago for us ,” but literally 13.5 billion years ago, well before conscious life existed, and thus at a time when there was no such thing as a correlate. Meillassoux also coins the term “ancestrality” (10) for the reality that predated the correlate, and later expands this term to “dia-chronicity,” (112) to refer to events occurring after the extinction of human beings no less than to those occurring before we existed. Up to this point, Meillassoux’s focus on ancestral entities existing prior to consciousness might seem like a straightforward realist who wants to unmask correlationism as just another form of idealism. Yet Meillassoux also admires the correlationist maneuver, which can obviously be traced back to Kant. Unlike a thinker such as Whitehead, Meillassoux feels no nostalgia for the pre-critical realism that came before Kant: “we cannot but be heirs of Kantianism,” he says (29). What impresses Meillassoux about correlationism is something both simple and familiar. If we attempt to think a tree outside thought, this is itself a thought . Any form of realism which thinks it can simply and directly address the world the way it is fails to escape the correlational circle, since the attempt to think something outside of thought is itself nothing other than a thought, and thereby collapses back into the very human-world correlate that it pretends to escape. For Meillassoux this step, suggested by Kant but first refined by the ensuing figures of German Idealism, marks decisive forward progress in the history of philosophy that must not be abolished. Any attempt to break free from the correlate must first acknowledge its mighty intellectual power. Realist though he may seem, Meillassoux’s works are filled with praise of such figures as Fichte and Hegel, not of so-called “naïve realists.” It is also the case that for Meillassoux, not all correlationisms are the same. The second pillar of his book is a distinction between various positions that I have termed “Meillassoux’s Spectrum,” though of course he is never so immodest as to name it after himself. He distinguishes between at least six different possible positions, and perhaps we could add even subtler variations if we wished. But in its simplest form, Meillassoux’s Spectrum allows for just four basic outlooks on the question of realism vs. anti-realism. Three of these are easy to understand, since we have already been discussing them. At one extreme is so-called “naïve realism,” which holds that a world exists outside the mind, and that we can know this world. Meillassoux rejects this naïve realism as having been overthrown by Kant’s critical philosophy. At the other extreme is subjective idealism, in which nothing exists outside the mind. For to think a dog outside thought immediately turns it into a thought, and therefore there cannot be anything outside; the very notion is meaningless. In between these two is what we have called correlationism. And here comes a crucial moment for Meillassoux, since he distinguishes between the two forms of “weak” and “strong” correlationism, and chooses the strong form as the launching pad for his own philosophy. Weak correlationism is easy to explain, since we all know it from the philosophy of Kant. The things-in-themselves can be thought but not known. They certainly must exist, since there cannot be appearances without something that appears. And we can think about them, which idealism holds to be impossible. They are simply unknowable due to the finitude of human thought. Strong correlationism is the new position introduced by Meillassoux (though he sees it at work in numerous twentieth century thinkers), midway between weak correlationism and subjective idealism. The major difference between the three positions is as follows. Weak correlationism says: “The things-in-themselves exist, but we cannot know them.” The subjective idealist says: “This is a contradiction in terms, since when we think the things-in-themselves, we already turn them into thoughts.” But the strong correlationist says: “Just because ‘things-in-themselves’ is a meaningless notion does not mean that they cannot exist. No one has ever traveled to the world-in-itself and come back to make a report on it. Thus, the fact that we cannot think things-in-themselves without contradiction does not prove that they do not exist anyway. There may be things-in-themselves, we simply are not capable of thinking them without contradiction form within the correlational circle.” This step is crucial for Meillassoux, since strong correlationism is the position he attempts to radicalize into his own new standpoint: speculative materialism. As I see it, this step of the argument fails. Strong correlationism cannot avoid collapsing into subjective idealism, since the statements of the strong correlationist are rendered meaningless from within. All three of the other positions in the Spectrum make perfectly good sense even for those who disagree with them. The naïve realist says that things-in-themselves exist and we can know them; the meaning of this statement is clear. The weak correlationist can say that things-in-themselves exist but lie forever beyond our grasp; this too makes perfect sense, even though the German Idealists try to show a contradiction at work here. We can also understand the claim of the subjective idealist that to think anything outside thought turns it into a thought, and that for this reason we cannot think the unthought. The strong correlationist, alone among the four, speaks nonsense . This person says “I cannot think the unthought without turning it into a thought, and yet the unthought might exist anyway.” But notice that the final phrase “the unthought might exist anyway” is fruitless for this purpose. For we have already heard that to think any unthought turns it into a thought. But now the strong correlationist wants to do two incompatible things simultaneously with this unthought. On the one hand, he neutralizes the unthought by showing that it instantly changes into just another thought. But on the other hand, he wants to appeal to the unthought as a haunting residue that might exist outside thought, thereby undercutting the absolute status of the human-world correlate found in idealism. But this is impossible. If you accept the argument that thinking the unthought turns it into a thought, you cannot also add “but maybe there is something outside that prevents this conversion from being absolutely true,” because this “something outside” is immediately converted into nothing but a thought for us. In short, Meillassoux here seems to be offering a kind of Zen koan: his “strong correlationism” is reminiscent of the gateless gate, or the sound of one hand clapping, or the command to punch Hegel in the jaw when meeting him on the road. We cannot at the same time both destroy the realist challenge of the things-in-themselves in order to undercut realism and reintroduce that very realist sense in order to undercut idealism. In a world where everything is instantly converted into thought, we cannot claim that there might be something extra-mental anyway, because this “might be something” is itself converted into a thought by the same rules that condemned dogs, trees, and houses to the idealist prison. This brings us to the third pillar of Meillassoux’s argument, which is the key to all the rest: the necessity of contingency. His strategy is to transform our supposed ignorance of things-in-themselves into an absolute knowledge that they exist without reason, and that the laws of nature can change at any time for no reason at all. In this way the cautious agnosticism of Kantian philosophies is avoided, but so is the collapse of reality into thought as found in German Idealism. Meillassoux does try to prove the existence of things-in-themselves existing outside thought; he simply holds that they must be proven after passing through the rigors of the correlationist challenge, not just arbitrarily decreed to exist in the manner of naïve realism. As he puts it, “Everything could actually collapse: from trees to stars, from stars to laws, from physical laws to logical laws; and this not by virtue of some superior law whereby everything is destined to perish, but by virtue of the absence of any superior law capable of preserving anything, no matter what, from perishing” (53). If idealism thinks that the human-world correlate is absolute, for Meillassoux it is the facticity of the correlate that is absolute. He tries to show this with a nice brief dialogue between five separate characters (55-59) which is covered in detail in my forthcoming book,2 but which I will simplify here for reasons of time. In this simplified version, we first imagine a dogmatic realist arguing with a dogmatic idealist. The realist says that we can know the truth about the things-in-themselves; the idealist counters that we can only the truth about thought, since all statements about reality must be turned into statements concerning our thoughts about reality. Here the correlationist enters and proclaims that both of these positions are equally dogmatic. For although we have access to nothing but thoughts, we cannot be sure that these thoughts are all that exist; there could be a reality outside thought, there is simply no way to know for sure. And this latter position is the one that Meillassoux attempts to transform from an agnostic, skeptical point into an ontological claim about the contingency of everything. Consider it this way. How does the correlationist defeat the idealist? The idealist holds that the existence of anything outside thought is impossible. The correlationist, by contrast, holds that something might exist outside the human-world correlate. But this “something might” has to be an absolute possibility. It cannot mean that “something outside thought might exist for thought ,” because that is what the idealist already says. No, the correlationist must mean that something might exist outside thought quite independently of thought. In other words, the correlationist says that idealism might be wrong, and this means it is absolutely true that idealism might be wrong. Thus, correlationism is no longer just a skeptical position. It holds that all the possibilities of the world are absolute possibilities. We have absolute knowledge that any of the possibilities about the existence or non-existence of things-in-themselves might be true, and this means that correlationism flips into Meillassoux’s own position: speculative materialism. As Meillassoux sees it, there are only two options here. Option A is to absolutize the human-world correlate, which is what the idealist does: there absolutely cannot be anything outside thought. Option B, by contrast, is to absolutize the facticity of the correlate: its character of simply being given to us, without any inherent necessity. The correlationist cannot have it both ways by saying: “there absolutely might be something outside thought, yet maybe this is absolutely impossible.” In other words, once we escape dogmatism we can only be idealists or speculative materialists, not correlationists. The human-world correlate is merely a fact, not an absolute necessity. But this facticity itself cannot be merely factical: it must be absolute. Here Meillassoux coins the French neologism factualité , which has been suitably translated into the English neologism “factiality.” (7, 122-3) Factialty means that for everything that exists, it is absolutely possible that it might be otherwise, not just that we cannot know whether or not it might be otherwise. Just as Kant transformed philosophy into a meditation on the categories governing human finitude, Meillassoux wishes to turn philosophy into a meditation on the necessary conditions of factiality, which he calls “figures”—a new technical term for him. (80) One such figure is that the law of non-contradiction must be true, and for an unusual reason. Since everything is proven to be contingent, nothing that exists can be contradictory, for whatever is contradictory has no opposite into which it might be transformed, and thus contingency would be impossible.3 Another such figure is that there must be something rather than nothing: for since contingency exists, something must exist in order to be contingent. It is a daring high-wire act, one that sacrifices realism to the correlational circle in order to rebuild it from out of its own ashes. Some might conclude that the lack of reason in things is a byproduct of the ignorance of finite humans, Meillassoux is making precisely the opposite point. For in fact, the doctrine of finitude usually leads directly to belief in a hidden reason. The fact that it lies beyond human comprehension merely increases our belief in this arbitrarily chosen concealed ground. By defending anew the concept of absolute knowledge Meillassoux evacuates the world of everything hidden. The reason for things having no reason is not that the reason is hidden, but that no reason exists. Thus, even while insisting on the necessity of non-contradiction, he rejects the other Leibnizian principle: sufficient reason. Everything simply is what it is, in purely immanent form, without deeply hidden causes. Or as Meillassoux puts it: “There is nothing beneath or beyond the manifest gratuitousness of the given—nothing but the limitless and lawless power of its destruction, emergence, or persistence” (63).The world is a “hyper-chaos”(64). But this is not the same thing as flux. For the chaos of the world is such that stability might occur just as easily as constant, turbulent change. Let’s now digress a bit, and return to the question of ancestrality, which Meillassoux transforms later in the book into “dia-chronicity.” Correlationism holds that all talk of a world outside the correlate is immediately recuperated by the correlate. The phrase “13.5 billion years ago” becomes “13.5 billion years ago for us ,” and the phrase “the universe following the extinction of humans” becomes “the universe following the extinction of humans for humans .” But notice that whether we talk about the world before or after humans, in both cases it is time that is used to challenge the correlate. Meillassoux has no interest in challenges that might be posed by space. For example, what about a vase in a lonely country house that topples to the floor and smashes when no one is there to watch it? Isn’t this also a challenge to correlationism, no less than the Big Bang or the heat death of the universe long after humans have vanished? In an eight-page supplement to the English translation of After Finitude ,4 possibly in response to my own 2007 review of the French original,5 Meillassoux bluntly denies that space is of any relevance to the question. Spatial distance is a merely harmless challenge to the human-world correlate. After all, even though no one is there in the lonely country house to witness the shattering of the vase, we can say that had there been an observer , that observer would have witnessed the toppling and destruction of the vase. For this event still occurs in a world in which the human-world correlate already exists, whereas the diachronicity of events both before and after the existence of humans makes it impossible to say that had there been an observer they would have witnessed the Big Bang occurring in such and such a fashion. However, it seems to me that Meillassoux merely asserts that the temporal simultaneity of our existence with that of the vase in the lonely country house is enough to render it harmless. It is true that the house does not exist prior to the correlate, but nonetheless it exists outside the correlate, and that is enough to make the same challenge. It is difficult to see why the “had there been an observer” maneuver succeeds in the case of a vase in the countryside in April 2011 but fails in the case of the Big Bang. This is not just a matter of nitpicking Meillassoux’s argumentative style: the fact that he bases his argument on time has at least two important consequences for his position. For in the first place, even though Meillassoux insists that the laws of nature are absolutely contingent, this turns out to be true only in a temporal sense. That is to say, it is a paradoxical feature of Meillassoux’s philosophy that he does allow for the existence of laws of nature, and simply believes that they can change at any moment without reason. Within any given moment, laws of nature do exist. He never suggests that different parts of the universe can have different laws at the same time, nor does he have any interest in the laws of part/whole composition that take place within any given instant. Could it be the case that rather than being made of gold atoms, a small chunk of gold could be made of silver atoms, cotton, horses, or that this same small piece of gold could be made of gigantic vaults filled with even more gold? These are not topics that draw Meillassoux’s attention, since he is focused solely on how the laws of nature might change or endure from one moment to the next . Another implication for Meillassoux’s system is that his concept of things-in-themselves turns out to be to be inadequate. For when he proves that things-in-themselves can exist without humans, this turns out to be true only in a temporal sense as well. Namely, things-in-themselves existed ten billion years ago, and they will continue to exist after all humans have succeeded in exterminating themselves. However, being able to exist before our births and after our deaths is just one small part of what it means to be a thing-in-itself. The more important part is that even if a thing is sitting on a table right now, in front of me, even if I stroke it lovingly or press my face up against it directly, I am still dealing only with a phenomenal version of the thing; the thing-in-itself continues to withdraw from all access. Yet no such thing is acknowledged by Meillassoux. For him finitude is a disaster, and absolute knowledge is in fact possible. Meillassoux’s thing-in-itself exists in independence only of the human lifespan , not of human knowledge. The fifth pillar of Meillassoux’s argument is his use of Cantor’s transfinite mathematics to show that even if the laws of nature are contingent, they need not be unstable, and thus we cannot use the apparent stability of nature to disprove his metaphysics of absolute contingency. What Cantor showed is that there are different sizes of infinity, and that all these infinities cannot be totalized in a single infinite number of infinities. Meillassoux sees this as crucial, since it allows him to discredit any “probabilistic” argument against his theory. The probabilistic argument (as defended quite clearly by Jean-René Vernes)6 would say this: given that the laws of nature seem so stable, it it is extremely improbable that there is no hidden reason for their remaining so stable. As Meillassoux sees it, probability is of value only when we can index an accessible total of cases. These can even be infinite: for example, there are an infinite number of points where a rope can break when stretched tight, but this does not stop us from calculating probabilities for various sections of the rope to break. By contrast, there is no way to sum up the number of possible laws of nature. For here there is no way to totalize; we cannot stand outside of nature and calculate the possible number of laws so as to determine the probabilities that any one of them might change. Therefore, although we can speak of probability when dealing with intraworldly events such as elections, horse races, and coin-flips, we cannot use the words “probable” or “improbable” when describing alterations at the level of nature as a whole. Rather than commenting on the validity of this argument and its use of Cantor, let me simply note that it once again creates a dualistic ontology. We already saw that Meillassoux treats time differently from space. In analogous fashion, he now treats the level of world differently from that of intraworldly events. The emergence of worlds is purely contingent and virtual and governed by no probability at all, while events within the world necessarily follow laws (even if these laws can change at any moment without reason), and thus their probabilities can be calculated. It is a strategy deeply reminiscent of Badiou’s (2005) own dualism between the normal “state of the situation” and the rare and intermittent “event.” The sixth and final pillar of Meillassoux’s book can be dealt with briefly, since we have already touched on it elsewhere. It comes at the very beginning of the book, when Meillassoux says that we must revive the distinction between primary and secondary qualities, and that the primary qualities are the ones that can be mathematized. He admits that he has not yet published a proof of this idea, though in fact it is already known as one of his primary doctrines. And here we encounter the familiar problem with Meillassoux’s inadequate conception of things-in-themselves. “Primary qualities” refers to those qualities that a thing has independently of its relations with us or anything else. But if the primary qualities can be mathematized, this means that they are not entirely independent of us, since our knowledge can get right to the bottom of them. The mathematized qualities of things are independent of us only in Meillassoux’s sense that they will still have those qualities even when all humans are dead. But to repeat, autonomy from the human lifespan is not the same as autonomy from human access. Here once more Meillassoux is concerned only with independence from the human-world correlate across time, not in any given instant. 2. L’Inexistence divine In 1997, the same year in which he turned thirty years old, Meillassoux earned his doctorate at the École normale supériuere with a brazen dissertation entitled L’Inexistence divine ( The Divine Inexistence ). The work was substantially revised in 2003. But even then, with typical fastidiousness, Meillassoux decided that the work was not yet ready for press. It has now been scrapped in favor of some future, multi-volume work bearing the same title. While writing my book on Meillassoux for Edinburgh University Press, I was permitted to translate excerpts from this unpublished work for use as an appendix in my own book; in total, the appendix contains approximately twenty percent of Meillassoux’s 2003 manuscript, the first time any of it will be published in any language. Nonetheless, a portion of the argument was already tested in the article “Spectral Dilemma,” published in English in the journal Collapse (2008: 261-75). There the philosophical motives for the virtual God are already made clear. What troubles us most are early deaths, brutal deaths, deaths of especial injustice– the sorts of deaths in which the brutal twentieth century was so abundant. And here, neither the atheist nor the believer can help us. The atheist can offer nothing but a sad and cynical resignation when reflecting on the victims of these terrible crimes. The believer does little better, being unable to explain how God could have allowed such things to happen, due to the famous intractability of the problem of evil. The solution offered to this dilemma by Meillassoux is bold, and all the more so given that he emerges from such a deeply Leftist, materialist, and unreligious background. His solution is that God does not yet exist, and therefore is not blameworthy for these catastrophes. Given that everything is contingent in Meillassoux’s philosophy, this God and divine justice might never exist, but they can at least exist as an object of hope. Let’s begin by jumping to the end of L’Inexistence divine , where the alternatives are laid out so nicely. There are four basic attitudes that humans can have towards God, Meillassoux says. First, we can believe in God because he exists. This is the classical theist attitude, rejected for the simple reason that it would be amoral and blasphemous to believe in a God who allows children to be eaten by dogs, to use Dostoevsky’s example. Second, we can disbelieve in God because he does not exist: the classical atheist attitude. But this leads to sadness, cynicism, and a sneering contempt for the greatness of human capacity. The third option, rather more complex, is to disbelieve in God because he does exist: in other words, to exist in rebellion against God as the one who must be blamed for the evils of the earth. The examples here might range from Lucifer himself, to the more human figure of Captain Ahab in Melville’s Moby-Dick , to Werner Herzog’s even more recent catchphrase, “Every man for himself, and God against all.” That leaves only the fourth option: believing in God because he does not exist. Meillassoux closes his book by saying that the fourth option has now been tried (namely, in the course of his own book), and that now that all four have been specified, we must choose. The first reaction to this theory of the inexistent God will be laughter. Few readers will ever be literally convinced by it, and probably none will immediately be convinced by it. But if we ask ourselves why we laugh, the answer is because it sounds so improbable that an inexistent God might suddenly emerge and resurrect the dead. It obviously sounds more like a gullible theology than a rigorous piece of philosophical work. Yet two things need to be kept in mind. First, Meillassoux’s theories are hardly more unlikely than those of great philosophers of the past such as Plato, Plotinus, Avicenna, Malebranche, Spinoza, Leibniz, Nietzsche, or Whitehead. We read the great philosophers not because their systems are plausible in commonsense terms that can be measured by the laws of probability. Instead, we read them precisely because they shatter the existing framework of common sense and open up new window on the universe. Second, and even more importantly, Meillassoux has already rejected probability as a valid measuring stick in philosophy. Or rather, he accepts probability in the intra-worldly realm (where it is linked with potentiality), and rejects it at the level of the world itself (where potentiality is replaced with what he calls virtuality). The virtual God can appear at any moment for no reason at all, just as any other new configuration of laws of nature can appear: in a manner that the laws of probability cannot calculate. Responding to those who might ridicule the idea of a sudden emergence of God and a resurrection of the dead, Meillassoux cites Pascal, who asserts that the resurrection of the dead would be far less incredible than the fact that we were born in the first place. This shifts philosophy onto new ground. Rather than concerning ourselves with what is likely to happen in the world as we know it, we focus instead on the most important things that could happen. For this reason, the expected objection that a virtual God is no more likely to appear than a virtual unicorn or a virtual flying spaghetti monster misses the point. Unicorns and spaghetti monsters could also appear, just like any other non-contradictory thing. But these would just be novel bizarre entities among others, not the heralds of completely new worlds. For Meillassoux, the emergence of matter, life, and thought have been the three truly amazing advents of the world so far, each of them dependent on the advent(s) preceding them. As he sees it, there can be no greater intraworldly entity than the human beings who already exist, since nothing in the world is better than the absolute knowledge of which humans alone are capable. This means that the next great advent must be something that perfects human beings rather than superseding them. And this can only be the world of justice, in which the dead are resurrected and their horrible deaths partially cancelled (Meillassoux never considers the possibility of a God who would literally erase the pre-divine past so that it never happened at all). The only immortality worth having is an immortality of this life, not an existence in some ill-defined afterworld. Human existence, he holds, must always be governed by a “symbol” that gives us the “immanent and comprehensible inscription of values in a world.” And just as cosmic history made the three great contingent leaps of matter, life, and thought, with a leap to justice as the only one still to come, a similar structure occurs within human culture and its symbols, which consist so far of the cosmological, naturalistic, and historical symbols, with a “factial” symbol still to come. We can review each of these symbols briefly. The cosmological symbol refers to the ancient dualism between the terrestrial and celestial spheres. Here below everything is conflict, corruption, and decay; but in the heavens nothing is perishable, all movement is circular, and everything is arranged in mutual harmony. This symbol is ended by modern physics when Galileo discovers such blemishes as sunspots and craters on the moon, and when Newton integrates both celestial and terrestrial movement into a single gravitational law. Next comes the naturalistic or romantic symbol, in which perfection comes not from the sky but from nature itself. The world is filled with pretty flowers (Meillassoux claims that the ancients never discussed the beauty of flowers until Plotinus in the third century) and with living creatures naturally moved by pity, at least until society corrupts them. This symbol collapses in the face of reality as we know it, since pity is no more common than war, corruption, and violence. This brings us to the historical symbol, which only now is passing away. Bad things may happen, but history has an inner logic of its own, such that everything works out in the end. The ultimate form of the historical symbol is the economic symbol, whether in a Marxist or neo-liberal form. Just as the Marxist holds that the inner economic logic of the capitalists will inexorably lead them to self-destruction, the neo-liberal assumes that the sum total of individual selfish actions will lead, in the long run, to the greatest possible good. We worship the economy and let it guide history for us, just as the ancients worshipped celestial bodies and held them to be free from blemish. The final remaining symbol is the factial symbol, which Meillassoux hopes will now emerge. Factiality, we recall, is his term for the absolute contingency of everything that exists. Once we have grasped this absolute contingency, we are free to expect the dramatic advent of the coming fourth World: the world of justice, inaugurated by a virtual God and even mediated by a messianic human figure. There is the added feature, however, that this messiah must abandon all claims to special status once the messianic realm of justice is achieved. The messianic figure will then be no more special than any person on the street, since a reign of human equality will have arisen. Although this focus on human being might seem like a return to standard humanism, Meillassoux holds that human pre-eminence has never truly been maintained. Previously, humans have been treated as special only because they contemplate the Good, because they resemble their omnipotent creator, or because they happen to be the temporary victors in a cruel Darwinian death-match between millions of living species. For Meillassoux, by contrast, humans have value because they know the eternal. But it is not the eternal that is important, since this merely represents the blind, anonymous contingency of each thing. What is important is not knowledge of the eternal , but knowledge of the eternal. We should not admire Prometheus for stealing fire from the gods; Prometheus is simply as bad as all the gods, no matter how much he increased our power. Feuerbach and Marx were wrong to say that God is a projection of the human essence, since for Meillassoux the usual concept of God represents the degradation of the human essence. If the traditional God was allowed to inflict plagues and tsunamis on the human race, the Promethean human of the twentieth century simply assumes the right to inflict death camps and atomic fireballs instead. In this respect, we have simply begun to imitate the degradation of humanity that was formerly invested in an omnipotent and arbitrary God. In response to charges that absolute contingency might lead to political quietism, Meillassoux counters that the World of justice would mean nothing unless we had already hoped for it beforehand. A World of justice that came along at random would merely be an improved third World of thought: indeed, a perfect one. But it would have satisfied no craving, and would therefore have no redemptive power. For this reason, we must actively hope for the fourth World of justice for such a fourth World ever to arise. Not only justice, but beauty is dependent on such hope: for Meillassoux, who is here somewhat dependent on Kant, beauty means an accord between our human symbolization and the actual world, which could never be present in a World of the blessed any more than justice could. And just as a messianic figure is needed to incarnate our hope and then abandon power once the World of justice is realized, it is the figure of the child whose fragile contingency shows us a dignity and a demand for justice beyond all power. 3. Meillassoux Radicalized or Reversed Given the promising reception of Meillassoux’s first book, it would not be groundless to engage in early speculation about what it might take to earn him a place in the history of philosophy. Maybe this will never happen—who knows?—but quite possibly it will: his lucid argumentative methods and sheer philosophical imagination at least make him a good candidate to be read well into the future, especially following further elaboration in print of his mature system. Philosophy is often practiced as thought it were nothing more than the amassing of “knockdown arguments.” But this is no more insightful than saying that good architecture is the amassing of steal beams. It is true that poorly constructed building cannot stand for long, but sound construction is merely the first, indispensable step in building. In fact, I am inclined to say that what really makes a philosopher important is not being right, but being wrong . I mean this in a very specific sense. I once heard the interesting remark about twentieth century culture that “you have to remember that the sixties really happened in the seventies.” That is to say, it was in the 1970’s rather than the more honored 1960’s that civil rights, free love, long hair, and the rock and roll drug culture really took root. With respect to the history of philosophy, we might just as easily say: “you have to remember that Plato really happened in Aristotle,” that “Kant really happened in Hegel” or “Hume really happened in Kant,” or that “Husserl’s phenomenology first achieved its truth in Heidegger.” One becomes an important philosopher not by being right, but by attracting rebellious admirers who tell you that you are wrong , even as their own careers silently orbit around your own. To recruit faithful disciples may be comforting and flattering, but the greatest thinkers have generally had to experience refutation at the hands of their most talented heirs. For this reason, I would propose that we size up the magnitude of living thinkers not by deciding how many times they are right and wrong, but by asking instead: who would take the trouble to refute this author? For this reason I do not ask: “Is Meillassoux right?”, since I do not believe in the virtual God myself, nor am I convinced by any important aspect of Meillassoux’s philosophy. Instead, I ask if there are interesting ways to overturn him. Only by being overturned, by no longer remaining a contemporary, does one become a classic. Let’s begin with a simple model of refutation, which can be refined further at a later date once the basic point is established. One kind of refutation simply consists in saying: “This author is a complete idiot.” The refuter now walks away in celebration, and no link between the present and the future is built; all is reduced to rubble. But this sort of mediocre triumphalism is generally practiced by those who achieve little of their own, and is not especially interesting. Much more interesting is the sort of refutation that does not take its target to be a complete idiot. I would like to suggest that there are just two basic ways in which this can be done: radicalization and reversal . It has not escaped my notice that this is a fairly good match for the Deleuzian distinction between irony and humor. Whereas irony critiques and adopts the opposite principle of what it attacks, humor accepts what it confronts but pushes it into highly exaggerated form. The ironist is like the worker who sows chaos by rebeling and contradicting the boss, while the humorist is like the worker who follows orders to an absurdly literal degree, with equally chaotic results. Let’s start with a few examples. In Aristotle’s treatment of Plato, and Heidegger’s of Husserl, we find reversal. Plato’s eidei are transformed by Aristotle into mere secondary substances, and the individual worldly things despised by Plato become what is primary. For Husserl what is primary is whatever is present to consciousness, while for Heidegger this is precisely what is secondary, since the primary stuff of the world withdraws from any form of presence at all. As for radicalization, it is most easily found in the transformation of Kant by German Idealism: “Kant was right to wall off the things-in-themselves from human access, and simply should have realized that the thought of the Ding an sich is also a thought, and thereby the noumena are just special cases of the phenomena,” with much following from this discovery. It would also be easy to read Spinoza as a radicalizer of Descartes, and Berkeley and Hume as radicalized versions of Locke. Perhaps the distinction is now sufficiently clear. Admiring refutations are not those that say “Professor X is an idiot,” which is merely the flip side of the eager disciple’s fruitless “Professor X got everything right.” Instead, it will be some variant of one of the following two options: “Professor X is important, but got it backwards,” or “Professor X is important, but didn’t push things far enough.” In the history of philosophy these two latter cases have often been painful in purely human terms: Aristotle expresses sadness at refuting Plato, Kant is openly annoyed at Fichte, and Husserl feels betrayed and used by Heidegger. Rude handling from later figures almost seems to be the sine qua non of being a great philosopher. Now, it has already been claimed that Meillassoux is an emerging philosopher of the first importance, and by no less a figure than Alain Badiou: “It would be no exaggeration to say that Quentin Meillassoux has opened up a new path in the history of philosophy…” (Preface, vii). But rather than taking Badiou’s word for it, or rejecting his word, we might experiment by asking how Meillassoux could be radicalized or reversed. Are there interesting ways of doing this that might launch whole new schools of philosophy, unexpected or even condemned by Meillassoux himself? While no one can see the future, the present is poor when it is not riddled with virtual futures. The relation between philosophers and their predecessors and successors is always somewhat complicated, of course. But generally there is one central divergence at stake, which might be taken as the key to all the others. On this basis we could say that new thinkers primarily radicalize or primarily reverse the main ideas of their chief philosophical forerunner. There may be specific historical conditions and perhaps even personality traits connected with these two types, but this question can be left aside for now. More important for us is that radicalizers will generally be followed by reversers, and vice versa. Consider the textbook example of a reversal in the history of philosophy: Kant’s Copernican Revolution, which inverts the so-called dogmatic tradition that addresses the world itself, and makes the world revolve instead around the conditions by which it is known. While it is not completely impossible that Kant’s successors might have re-reversed this principle back into a new and stronger dogmatic realism, conditions were premature for such a move. Anyone doing this too early would likely have been an angry anti-Kantian reactionary rather than an original thinker in command of a genuinely new realist principle. The far more likely outcome is the one that actually happened: Kant’s reversal of his predecessors was viewed as incomplete, or as retaining lamentable bits of the traditional view, which despite his admirable breakthrough he was unable to shake off. This was the view of German Idealism, anyway. In similar fashion, Spinoza could also be viewed as a radicalizer of Descartes, who is equally accused of preserving various Scholastic dogmas in an otherwise radical project of philosophical reversal. The point is this: reversals in the history of thought tend to be followed soon thereafter by radicalizations of those reversals. The same may hold true in reverse: radicalizations might generally be followed by reversals, given that it is not always possible to be more radical than the radicals have already been. Consider the case of Husserl, who radicalizes Brentano’s early vagueness about what lies beyond immanent objectivity, and Twardowski’s assertion that there must be an external object lying outside the intentional content, by collapsing everything into the intentional sphere: there is no difference between the Berlin in my consciousness and the actual Berlin that is home to millions of people. It is difficult to see how one could be even more radical than Husserl’s idealist turn here. And thus the road is paved to Heidegger’s reversal of classical phenomenology, in which the key point is what lies deeper than any presence to consciousness: the Sein whose power and obscurity cannot be made exhaustively present, but only sends itself in historical epochs. In similar fashion we might also read Leibniz as a reverser of Spinoza’s radicalization, retrieving a strong sense of individual substance and a certain validity of what the Scholastics had said. Returning to Meillassoux, we might ask which kind of philosopher he is: a radicalizer or a reverser? At present, Meillassoux looks to me like a radicalizer (though for now his future remains shrouded in mist). He takes the correlationist tradition, which allows us to speak only of the relation between human and world, and tries to raise it into an even more extreme claim about the absolute contingency of everything. But whereas German Idealism did this by trying to collapse the distinction between thought and world entirely into the “thought” side, Meillassoux does it by trying to shift the non-absolute contingency of the thought-world correlate from epistemology to ontology. It is no longer a question of the inability of human knowledge to know what lies outside the correlate, but the inability of reality itself to be rooted in any definite laws. Furthermore, if we look at the various features of Meillassoux’s philosophy identified earlier tonight, all but one are already so radical that there is no obvious way to push them further. The one exception would be his claim that the world as a whole can change for no reason at any moment, coupled with the inconsistent claim that within a given world there are laws of nature that everything must follow. If gravitational attraction between all masses is a current law in our world, then for Meillassoux there can be no exceptions to this law for as long as it remains in force. A toppled vase will fall to the floor every time for sure,unless there is a cosmic change by which the laws of nature as a whole have altered. (This is reminiscent of the late medieval distinction between the absolute and ordained power of God, according to which God has the power to set or change the laws of nature, but not to contravene those laws locally once they are set.) On this point, to radicalize Meillassoux would simply be to say: there are no laws of nature even in the local sense. Everything that happens, even in the world here and now, is purely contingent and not governed by even a trace of law. And while this would be a more consistent development of Meillassoux’s thoughts on contingency, it is difficult to see how it could lead to a new philosophy. Instead, the admiring successors of Meillassoux are more likely to reverse one of his already sufficiently radical points. At least four candidates come to mind: *First, we have seen that Meillassoux thinks correlationism is challenged by a time before or after consciousness, but not by a space lying outside it. Perhaps this could be reversed into saying that spatial exteriority is the really crucial point. The arguments on this point are perhaps the least convincing in After Finitude (and do not even occur in the original French edition), and therefore it might be a candidate for the “blind spot” of which no philosopher is ever free. *Second, Meillassoux uses Cantor to claim that the contingency of laws of nature would not entail that they are unstable. A successor of Meillassoux might claim that it does make them unstable, and celebrate this fact. This person would then have to explain why common sense seems to encounter a relatively stable world despite its truly rampant instability. Whereas Meillassoux’s problem is to show how stability might exist despite contingency, this successor’s problem would be slightly different: to show why actual, full-blown instability might have the appearance of stability. *Third, Meillassoux claims that the primary qualities of things are those that can be mathematized. He might be reversed by a successor who says the opposite: the mathematizable qualities are the secondary ones, and the primary ones are those that elude symbolic formulation. While this is a perfectly valid possible objection to Meillassoux, it is one that is made in advance by some of his predecessors and is still made by some of his peers, making it less interesting for futurology than some of his other points. *Fourth and finally, whereas Meillassoux claims that God does not exist but might exist in the future, a successor might argue even more bizarrely that God has always existed but might vanish in the future. Let’s arbitrarily select the first of these possibilities, and imagine briefly where it might lead, if pursued in the future by admiring detractors. Meillassoux comes from the circle of Badiou, and some of Badiou’s most ardent admirers are found in Latin America. So, let’s imagine that towards mid-century some ingenious reversers of Meillassoux emerge in that portion of the Spanish-speaking world. Just for fun, let’s call them Castro and Chávez. And in order to avoid any confusion with the present-day politicians of those names, we will stipulate that Meillassoux’s great successors are both women. The philosopher Castro (we will suppose she comes from Peru) reverses Meillassoux’s argument that the ancestral or diachronic are what most threaten the human-world correlate. Instead, she claims that the diachronic does not threaten the correlate at all, and that we must instead look at space as what ruins the correlate and demands a strange new realism. What would such a philosophy look like? In order to determine this, we might ask what price Meillassoux pays for doing it the opposite way. As I see it, he pays in two separate ways. One is that laws of nature for him are contingent over time . The laws of nature apply to the universe as a whole at any given moment, and would be changed globally if they are ever changed at all. The second price he pays is that Meillassoux has no mereology , or theory of parts and wholes. Everything for him is on the level of the given, or immanent in experience, with the sole proviso that the laws governing this immanence might change without notice at any given moment. In reversing Meillassoux, Castro makes the following claims in the preface to her stunning debut book of 2045, The Cosmos and its Neighborhoods , rapidly translated from Spanish into all the languages of the world: Despite his brilliant analysis of the contingency of laws of nature over time, Meillassoux gets two important assumptions wrong. First, he allows for only one set of contingent laws to govern nature as a whole. Second, he allows laws to govern only the world that is immanent in experience, and thereby fails to explore the contingency among part-whole relations. In this book I will argue, first, that the laws of nature vary in any given instant between one region of the universe and the next; and second, that the world is made up of layers of parts and wholes that are also contingent with respect to one another. Those are the words of Castro. This may sound like a hopeless free-for-all of chaos, yet the book somehow succeeds in drawing some compelling deductions about how laws must vary from one place or level in the world to the next. Trapped in the limited horizon of 2011, and not yet inspired by the heavily balkanized political and technological situation of 2050 that somehow lends additional credence to Castro’s vision, we can only vaguely grasp what such a philosophy might look like. After this reversal of Meillassoux by Castro, the usual pattern leads us to expect a radicalization by Chávez, a young Argentine student of Castro. How could the already strange theories of Castro be radicalized? Perhaps as follows, in a disturbing new book entitled The Implosion of the Neighborhoods , which argues as follows: Castro was right to shift the Meillassouxian framework of contingency from time to space. However, in this respect she retained a surprisingly traditional opposition between the two. In this book I will show that time and space collapse into one another. This may sound too much like the discredited four-dimensional block universe of twentieth century physics and philosophy. However, the four-dimensional universe is a model biased in favor of space, merely adding an extra dimension to the commonsense spatial continuum while stipulating that the serial passage of time is an illusion. In this book I will argue instead for a one-dimensional space-time modeled after our experience of time, in which there is no simultaneous co-existence at all between different parts of the universe, or ‘neighborhoods’ as my esteemed teacher Castro has called them. Instead, the various portions of the universe link to one another by succession rather than by coexistence. Buenos Aires, New York, and Amsterdam do not exist simultaneously in the same landscape, but one after the other in the mind of some observer, and this observer can only be an observer much larger than any human. Against Meillassoux’s notion of a virtual God that does not exist now but might exist in the future, I will argue for an actual God that surveys the universe in sequence, thereby generating the illusion of spatial diversity and even the illusion of individual minds located within that diversity. Once this divine observer dies, the universe as a whole must perish. Again, these ideas are so bizarre that we of 2011 can barely comprehend them, just as Aristotle would have had a difficult time grasping the theories of Descartes. We could then perhaps imagine a further reversal of this theory, emanating from the intellectually resurgent Philippines of the twenty-second century. The Filipino School might argue that the universe is already dead, given the collapse of its spatial richness into the serial observations of a flimsy and mortal God. The virtual universe does not yet exist, but might exist in fully spatial form in the future, and this would require the death of God and the resulting liberation of God’s succession of images as independent, spatially situated realities. With a bit of sharpening, we might be able to make all of these imaginary thinkers more intuitively clear. Along with the history of philosophy, there might arise a new discipline generating imaginary futures for philosophy. The richness of Meillassoux’s system comes not from the fact that he is plausibly right about so many things, but because his philosophy offers such a treasury of bold statements ripe for being radicalized or reversed. He is a rich target for many still-unborn intellectual heirs, and this is what gives him the chance to be an important figure. NOTES 1. Quentin Meillassoux, After Finitude . Trans. R Brassier. (London: Continuum, 2008.) Page 5. The word “correlationism” does not appear in his doctoral thesis. As Meillassoux informed me in an email of February 8, 2011, he first coined this term in 2003 or 2004, while editing for publication a lecture he had given at the École normale supérieure on a day devoted to the theme of “Philosophy and Mathematics,” an event including Alain Badiou as one of the participants. 2. Graham Harman, Quentin Meillassoux: Philosophy in the Making . (Edinburgh: Edinburgh University Press, forthcoming 2011). 3. In an email of December 6, 2010, Meillassoux clarifies that in After Finitude he only deduces the impossibility of a “universal contradiction,” not of a determinate contradiction. In the same email he suggests that he can also prove the latter, though the proof is somewhat lengthier than the one found in After Finitude . 4. After Finitude (18-26), in the passage falling between the two sets of triple asterisks. These pages were sent by Meillassoux to translator Ray Brassier (in French) during the translation process, and do not appear in the original French version of the book. 5. Graham Harman, “Quentin Meillassoux: A New French Philosopher.” Philosophy Today 51.1 (2007): Pages 104-117. The passage where I raise the question of space can be found in the first column of page 107. 6. Vernes is first cited on p. 95 of After Finitude . See Jean-René Vernes, Critique de la raison aléatoire, ou Descartes contra Kant . (Paris: Aubier, 1982).  . (shrink)

Traditional theistic arguments conclude that God exists. Pragmatic theistic arguments, by contrast, conclude that you ought to believe in God. The two most famous pragmatic theistic arguments are put forth by Blaise Pascal (1662) and William James (1896). Pragmatic arguments for theism can be summarized as follows: believing in God has significant benefits, and these benefits aren’t available for the unbeliever. Thus, you should believe in, or ‘wager on’, God. This article distinguishes between various kinds of theistic wagers, including (...) class='Hi'>finite vs. infinite wagers, premortem vs. postmortem wagers, and doxastic vs. acceptance wagers. Then, we’ll turn to the epistemic-pragmatic distinction, and discusses the nuances of James’ argument, and how views like epistemic permissivism and epistemic consequentialism provide unique “hybrid” wagers. Finally, we’ll cover outstanding objections and responses. (shrink)

In this substantive book, Robert Adams distills and crystallizes much of his previous work into an impressive two-tiered ethical framework: a divine nature theory of the Good and a divine command theory of the morally obligatory. The result is an expansive, integrated, and sophisticated ethical theory that merits great attention. Four major parts comprise the book: The Nature of the Good, Loving the Good, The Good and the Right, and The Epistemology of Value.

This contribution sheds light on the role of infinite idealization in structural analysis, by exploring how infinite elements and finite element methods are combined in civil engineering models. This combination, I claim, should be read in terms of a ‘complementarity function’ through which the representational ideal of completeness is reached in engineering model-building. Taking a cue from Weisberg’s definition of multiple-model idealization, I highlight how infinite idealizations are primarily meant to contribute to the prediction of structural (...) behavior in Multiphysics approaches. (shrink)

This contribution sheds light on the role of infinite idealization in structural analysis, by exploring how infinite elements and finite element methods are combined in civil engineering models. This combination, I claim, should be read in terms of a ‘complementarity function’ through which the representational ideal of completeness is reached in engineering model-building. Taking a cue from Weisberg’s definition of multiple-model idealization, I highlight how infinite idealizations are primarily meant to contribute to the prediction of structural (...) behavior in Multiphysics approaches. (shrink)