Although the concept of ‘true infinity’ is undoubtedly central to Hegel's philosophy, the Anglophone rehabilitation of Hegel as a post-Kantian critical philosopher has avoided any sustained interpretive confrontation with the concept. In this paper, I provide a revisionary reconstruction of Hegelian true infinity by engaging with Martin Hägglund's argument in This Life (2019) for the centrality of finitude to Hegel's philosophy. For Hägglund, Hegel's philosophy effects a ‘secular reconciliation’ with finitude by demonstrating that our mortality is (...) not a negative condition to be overcome; rather, mortality is constitutive of rational social agency or ‘spiritual life’. While Hägglund's interpretive emphasis on finitude prima facie occludes a consideration of the concept of true infinity, I show that Hägglund's understanding of spiritual life in terms of the dynamic of self-maintenance implicitly articulates a revisionary understanding of true infinity as instantiated in and through the activity of finite living rational beings. More specifically, I show that Hägglund's understanding of finitude as constitutive of spiritual life is grounded in three closely interrelated Hegelian meta-concepts, all of which Hegel derives in the second chapter of the Logic (‘Existence’): (a) the distinction between abstract and determinate negation; (b) the form of individuality as constituted through determinate negation; and (c) true infinity as the form of individuality understood in explicitly processual terms. I thus ground a deflationary interpretation of true infinity in Hegel's logical account of determinate individuality, and at the same time, contribute to This Life's ongoing critical reception by articulating its Hegelian logical infrastructure. (shrink)

This article proposes that the foundations of Hegel’s contribution to social criticism are compatible with, and enriched by, his meta-theology. His social critique is grounded in his belief that normative ideas – and especially the idea of freedom – are necessarily experiential and historical. Often regarded as a recipe for an authoritarian reconciliation with the status quo, Hegel’s philosophy has been dismissed by some unsympathetic commentators from the left as inimical to the task of social criticism. Much of the reason (...) for this has been the opinion that his systematic approach to philosophy is one underpinned by either a highly unorthodox theism or, more commonly, pantheism. Even scholars who wish to defend Hegel as a social critic have tended to abandon or at least massively downplay his meta-theology. In this article, I argue that it is precisely the emphasis on his original theo-metaphysics that offers a powerful and relevant contribution to social criticism. Hegel then becomes an important contributor, from the tradition of social criticism, to the growing trend in academia and wider society of rethinking the relationship between the religious and the secular, known as post-secularism. The proposition at the centre of his system – that human history and society are necessary moments in the process of divine self-understanding – is not new. But the specifics of Hegel’s concept of God that I am proposing – and, moreover, their implications for his political thought – are new. I propose that Hegel was neither a theist nor a pantheist but, rather, a dialectical panentheist. According to panentheism, God is neither straightforwardly transcendent to nature and history nor immanently identifiable with nature and history, but rather is dynamically and dialectically immanent in the ongoing processes of self-transcendence that are nature and history. If such an interpretation of Hegel’s system is valid then the proposition that his political thought is an exercise in dialectical panentheistic theology is one that is worth making and defending because it has important things to say about his social critique. (shrink)

Many philosophic arguments concerned with infinite series depend on the mutual inconsistency of statements of the following five forms: (1) something exists which has R to something; (2) R is asymmetric; (3) R is transitive; (4) for any x which has R to something, there is something which has R to x; (5) only finitely many things are related by R. Such arguments are suspect if the two-place relation R in question involves any conceptual vagueness or inexactness. Traditional sorites arguments (...) show that a statement of form (4) can fail to be true even though it has no clear counter-example. Conceptual vagueness allows a finite series not to have any definite first member. I consider the speculative possibilities that there have been only finitely many non-overlapping hours although there has been no first hour and that space and time are only finitely divisible even though there are no smallest spatial or temporal intervals. (shrink)

We suggest a new approach for addressing the problem of establishing an axiomatic foundation for large cardinals. An axiom asserting the existence of a large cardinal can naturally be viewed as a strong Axiom of Infinity. However, it has not been clear on the basis of our knowledge of ω itself, or of generally agreed upon intuitions about the true nature of the mathematical universe, what the right strengthening of the Axiom of Infinity is—which large cardinals ought (...) to be derivable? It was shown in the 1960s by Lawvere that the existence of an infinite set is equivalent to the existence of a certain kind of structure-preserving transformation from V to itself, not isomorphic to the identity. We use Lawvere's transformation, rather than ω, as a starting point for a reasonably natural sequence of strengthenings and refinements, leading to a proposed strong Axiom of Infinity. A first refinement was discussed in later work by Trnková—Blass, showing that if the preservation properties of Lawvere's tranformation are strengthened to the point of requiring it to be an exact functor , such a transformation is provably equivalent to the existence of a measurable cardinal. We propose to push the preservation properties as far as possible, short of inconsistency. The resulting transformation V→V is strong enough to account for virtually all large cardinals, but is at the same time a natural generalization of an assertion about transformations V→V known to be equivalent to the Axiom of Infinity. (shrink)

This paper addresses what I call ‘the constitutive question’ concerning the rules we follow: namely, what determines the standard for a rule's correct application. John McDowell has offered a putative ‘middle position’ between two extreme, unacceptable answers: empirical idealism, which takes the requirements of a rule in any given situation to be constituted by our reaction to the case; and hard platonism, which takes these requirements to be delivered by unvarnished reality as absolutely the simplest or most natural way to (...) carry on. Tellingly, however, McDowell's position is itself unacceptably idealist in his picture of the way in which we are ‘involved’ on the right-hand side of biconditionals such as, ‘“Diamonds are hard” is true if and only if diamonds are hard’.In response to this stultifying state of affairs, I suggest that McDowell has followed the empirical idealist and hard platonist in assuming that the requirements of a rule must be grounded by something else: something external to the rule, which mediates between the rule and the standard for its correct application. This assumption is false. What is more, one application of a metaphor to which Wittgenstein has a somewhat equivocal relation—that of rules as rails invisibly laid to infinity—can help this point to stick. (shrink)

As José Benardete observes, "the concept of the infinite is found to impinge on almost the whole schedule of ontological questions" (Infinity, viii). This is especially true for the early moderns, for whom questions like the following were still very much in play: Does the world have a beginning? Are there bounds to the spatial extent of the world? How does an imperfect creation flow from an infinitely perfect creator? How does the infinite divisibility of the continuum relate (...) to the structure of the material world? How are human actions related to the (possibly) infinite chain of causes or influences that shape our behaviors? Far from being an esoteric topic, which is best relegated to the outskirts of metaphysics or mathematics, infinity has tendrils reaching into nearly every philosophical domain. (shrink)

"'There is only one thing that is certain, namely that we can have nothing certain; and therefore it is not certain that we can have nothing certain,' Samuel Butler once said, expressing in that mindbloggler all the elements required to form a classical paradox. Throughout the ages wise men and jesters alike have been intrigued by such mental twists and riddles which defy common sense and yet appear to be true." -- Dust jacket.

Why should some essential properties of geometry (i.e., infinity, symmetry, and dimensionality) be both necessary and desirable in the way that they have been constructed albeit with different modifications over time since time immemorial? Contrary to the conventional wisdom in all history hitherto existing, the essential properties of geometry do not have to be both necessary and desirable. This is not to suggest, of course, that one has nothing to learn from geometry. On the contrary, geometry has contributed to (...) the advancement of knowledge in many ways since its inception as a field of knowledge some millennia ago. The point in this book, however, is to show an alternative (better) way to understand the nature of geometry, which goes beyond human conception, intuition, and imagination, together with worldly experience of course, as its foundation, while learning from them all with theoretical implications for time travel, hyperspace, and other important issues. If true, this seminal view will fundamentally change the way that the nature of abstraction in the thinking process is to be understood, with its enormous implications for the future advancement of knowledge, in a small sense, and what I originally called its "post-human" fate, in a large one. (shrink)

We consider an approach to some philosophical problems that I call the Method of Conceptual Articulation: to recognize that a question may lack any determinate answer, and to re-engineer concepts so that the question acquires a definite answer in such a way as to serve the epistemic motivations behind the question. As a case study we examine “Galileo’s Paradox”, that the perfect square numbers seem to be at once as numerous as the whole numbers, by one-to-one correspondence, and yet less (...) numerous, being a proper subset. I argue that Cantor resolved this paradox by a method at least close to that proposed—not by discovering the true nature of cardinal number, but by articulating several useful and appealing extensions of number to the infinite. Galileo was right to suggest that the concept of relative size did not apply to the infinite, for the concept he possessed did not. Nor was Bolzano simply wrong to reject Hume’s Principle (that one-to-one correspondence implies equal number) in the infinitary case, in favor of Euclid’s Common Notion 5 (that the whole is greater than the part), for the concept of cardinal number (in the sense of “number of elements”) was not clearly defined for infinite collections. Order extension theorems now suggest that a theory of cardinality upholding Euclid’s principle instead of Hume’s is possible. Cantor’s refinements of number are not the only ones possible, and they appear to have been shaped by motivations and fruitfulness, for they evolved in discernible stages correlated with emerging applications and results. Galileo, Bolzano, and Cantor shared interests in the particulate analysis of the continuum and in physical applications. Cantor’s concepts proved fruitful for those pursuits. Finally, Gödel was mistaken to claim that Cantor’s concept of cardinality is forced on us; though Gödel gives an intuitively compelling argument, he ignores the fact that Euclid’s Common Notion is also intuitively compelling, and we are therefore forced to make a choice. The success of Cantor’s concept of cardinality lies not in its truth (for concepts are not true or false), nor its uniqueness (for it is not the only extension of number possible), but in its intuitive appeal, and most of all, its usefulness to the understanding. (shrink)

The present article critically examines three aspects of Graham Priest's dialetheic analysis of very important kinds of limitations (the limit of what can be expressed, described, conceived, known, or the limit of some operation or other). First, it is shown that Priest's considerations focusing on Hegel's account of the infinite cannot be sustained, mainly because Priest seems to rely on a too restrictive notion of object. Second, we discuss Priest's treatment of the paradoxes in Cantorian set-theory. It is shown that (...) Priest does not address the issue in full generality; rather, he relies on a reading of Cantor which implicitly attributes a very strong principle concerning quantification over arbitrary domains to Cantor. Third, the main piece of Priest's work, the so-called “inclosure schema”, is investigated. This schema is supposed to formalize the core of many well-known paradoxes. We claim, however, that formally the schema is not sound. (shrink)

The question whether quantum discontinuity can or cannot provide an answer to Zeno's Paradoxes is reopened. It is observed that what is usually understood by the term "discontinuity", namely, Einstein's conception of the photon as described by himself and all others, is unsuitable to the task because, essentially,it reduces to the trivial 'discontinuity' of objects scattered in space. By contrast, quantization of energy levels, which are not in space but can only alternate in time, provide the right sort of discontinuity (...) required. Discrete quantized orbits, corresponding to eigen-frequencies, are irreducible, and nothing is allowed to stand in-between them in satisfaction of the quantum postulate, furnishing the requisite, and so far missing, immediate nextness of a point to a certain other. In this way, Zeno's Runner need not postpone his first step indefinitely, always waiting upon an infinity of preceding steps, before it can be taken. There is now a point that is next to a point and so a step on that point, which is the first step. It follows that, if one kind of discontinuity, Einstein's, is incapable of offerring an answer to Zeno, while another kind can, the two are discrepant. One of them, the former, is not a kind of discontinuity properly so called at all, though evidently the consequence of one. (shrink)

The question whether quantum discontinuity can or cannot provide an answer to Zeno’s Paradoxes is reopened. It is observed that what is usually understood by the term “discontinuity”, namely, Einstein’s conception of the photon as described by himself and all others, is unsuitable to the task because, essentially,it reduces to the trivial ‘discontinuity’ of objects scattered in space. By contrast, quantization of energy levels, which are not in space but can only alternate in time, provide the right sort of discontinuity (...) required. Discrete quantized orbits, corresponding to eigen-frequencies, are irreducible, and nothing is allowed to stand in-between them in satisfaction of the quantum postulate, furnishing the requisite, and so far missing, immediate nextness of a point to a certain other. In this way, Zeno’s Runner need not postpone his first step indefinitely, always waiting upon an infinity of preceding steps, before it can be taken. There is now a point that is next to a point and so a step on that point, which is the first step. It follows that, if one kind of discontinuity, Einstein’s, is incapable of offerring an answer to Zeno, while another kind can, the two are discrepant. One of them, the former, is not a kind of discontinuity properly so called at all, though evidently the consequence of one. (shrink)

The question whether quantum discontinuity can or cannot provide an answer to Zeno’s Paradoxes is reopened. It is observed that what is usually understood by the term “discontinuity”, namely, Einstein’s conception of the photon as described by himself and all others, is unsuitable to the task because, essentially,it reduces to the trivial ‘discontinuity’ of objects scattered in space. By contrast, quantization of energy levels, which are not in space but can only alternate in time, provide the right sort of discontinuity (...) required. Discrete quantized orbits, corresponding to eigen-frequencies, are irreducible, and nothing is allowed to stand in-between them in satisfaction of the quantum postulate, furnishing the requisite, and so far missing, immediate nextness of a point to a certain other. In this way, Zeno’s Runner need not postpone his first step indefinitely, always waiting upon an infinity of preceding steps, before it can be taken. There is now a point that is next to a point and so a step on that point, which is the first step. It follows that, if one kind of discontinuity, Einstein’s, is incapable of offerring an answer to Zeno, while another kind can, the two are discrepant. One of them, the former, is not a kind of discontinuity properly so called at all, though evidently the consequence of one. (shrink)

In “Violence and Metaphysics” Jacques Derrida suggests that “the only effective position to take in order not to be enveloped by Hegel would seem to be…to consider false-infinity…irreducible.” Inversely, refuting the charge of logocentrism associated with Hegelian true infinity ( wahrhafte Unendlichkeit ) would involve showing that Hegel’s speculative logic does not establish the infinity of being exempt from the negativity of the finite. This paper takes up Derrida’s challenge, and argues that true infinity (...) is crucial to Hegel’s understanding of ideality as a question of normative authority, which does not fall prey to logocentrism. Through an exposition of the dialectic of the finite and the infinite in Hegel’s Science of Logic , I argue that true infinity is not an ontological category that eliminates division, but rather refers to the metalogical standpoint involved in a philosophical account of determinacy. Although fully achieved at the end of the Logic , the metalogical standpoint that Hegel elaborates in the Seinslogik under the banner of the true infinite already clarifies that determinacy is a product of normative authority that is itself precarious. (shrink)

On 27 February 1998, the Indian Institute of Management, Calcutta and the office of the Fulbright binational educational exchanges in Calcutta jointly hosted a seminar on 'The Tradition of Non Violence: The American Experience and the Gandhian' at the Management Centre for Human Values. There were two keynote presentations. The one on the American experience was by Michael True, Professor of English Literature at Assumption College, Massachusetts, who was teaching as Fulbright visiting lecturer at Utkal University, Bhubaneswar. The Gandhian (...) case was presented by Profes sor Amlan Datta, economist and writer, also former Vice-Chancellor of Visva-Bharati University, Santiniketan. The session was chaired by Professor Amitava Bose, Director, Indian Institute of Management. To close the discussions, a summing up was presented by Professor S.K. Chakraborty, Convenor, Management Centre for Human Values. The importance of discussions on the subject of non-violence and the Gandhian alterna tive cannot be overestimated in this year of the 50th anniversary of Indian independence and a nationwide soul-searching over these five decades. The US Educational Foundation in India contributed to the national discussion through this bilateral seminar and its comparative perspective. (shrink)

What does world peace mean? Peace is more than the absence and prevention of war, whether international or civil, yet most of our ways of conceptualizing and measuring peace amount to just that definition. In this essay, as part of the roundtable “World Peace,” I argue that any vision of world peace must grapple not only with war but with the continuums of violence and peace emphasized by feminists: running from the home and community to the public spaces of international (...) relations. Breaking free of the constraints of the last century's intellectual boundaries, I suggest that war and peace are not a dichotomy but rather are intimately related. Yet the dearth of feminist perspectives in global debates prevents us from seeing how violence and harm are exacerbated in households and through the global economy under conditions of both “war” and “peace.” To understand the possibilities for world peace, we must understand these varieties of violence and harm that threaten peace. And tosustainpeace we must address the harmful gendered identities, ideologies, and social dynamics that support violence in every society. A narrow understanding of peace as merely the absence of organized violence does not engender the kind of nuanced and rich understanding of human history and human relations needed to bring an end to the structural and physical violence that remains pervasive worldwide. (shrink)

This volume addresses the way ethics is taught in American Business Schools. The Editors has assembled a collection of timely essays offering practical experienced-based insights in business education. The authors of these essays address a diversity of topics yet are unanimous in calling for change (even if they occasionally disagree on the best means of accomplishing it). For business faculties seeking to meet this growing and multifaceted challenge within their discipline, this book offers a wealth of useful insights and practical (...) solutions. (shrink)

What are we to make of the New Christian Right’s loss of political influence and the rise of the Tea Party and libertarianism more broadly? Rather than imagine a coalition of resentment as does William E. Connolly, this paper argues that several key religious ideas of protestant fundamentalism have become secularized and now function as a political theology that privileges the personal and marginalizes the public arena. American fundamentalism shares several characteristics with protestant fundamentalism—even as it represents what might be (...) called a migration of the holy. (shrink)

After the re-emergence of the tradition of virtue ethics in the early 1980s Reinhold Niebuhr has often served as a foil for authors who locate themselves in that tradition. However, this exercise has often proved controversial. This collection of essays continues this work, across a wide range of subjects, with the aim of avoiding some of the polemics that have previously accompanied it. The central thesis of this book is that putting the work of Reinhold Niebuhr and Christian realism in (...) dialogue with contemporary virtue theory is a profitable undertaking. An introductory essay argues against locating Niebuhr as a consequentialist and in favour of thinking of his work in terms of a dispositional ethics Contributors take different positions on whether Niebuhr's dispositional ethics should be considered a form of virtue ethics or an alternative to virtue ethics. Several of the articles relate Niebuhr and Christian realism to particular virtues. Throughout there is an appreciation of the ways in which any Niebuhrian approach to dispositional ethics or virtue must be shaped by a sense of tragedy, paradox, or irony. The most moral disposition will be one which includes doubts about its own virtue. This volume allows for a repositioning of Niebuhr in the context of contemporary moral theory as well as a rereading of the tradition of virtue ethics in the light of a distinctly Protestant, Christian realist and paradoxical view of virtue. As a result, it will be of great interest to scholars of Niebuhr and Christian Ethics and scholars working in Moral Philosophy and the Philosophy of Religion more generally. (shrink)

La comparaison des portraits à l’huile réalisés par Romaine Brooks dans les années 20 et les photographies de Claude Cahun et sa partenaire Marcel Moore de la même époque rend visible les positions idéologiques conflictuelles que ces women of vision défendirent dans les milieux artistique et politique où les débats sur la femme moderne se sont élaborés.

This chapter contains sections titled: Hegel on Love: The Early Theological Writings Recognition and Spirit: Hegel's Appropriation and Critique of Fichte Hegel's Philosophical Theology: Love, Reconciliation, True Infinity.

Metaphysics and the Modern World makes the abiding questions of the nature of the self, world, and God available for the modern reader. Donald Phillip Verene presents these questions in both their systematic and historical dimensions, beginning with Aristotle's claim in his Metaphysics that philosophy begins in wonder. The first three chapters concern the origin of metaphysics as the transformation of the conception of reality in ancient Greek mythology, the ontological argument as the basis of Christian metaphysics, and the Renaissance (...) cosmology of infinite worlds and the coincidence of contraries. The final four chapters present the central issues of the metaphysics of history through the New Science of Vico, the principle of true infinity of Hegel's Logic, the dialectic of spirit and life in Cassirer's Metaphysics of Symbolic Forms, and the conception of actual entities and God in Whitehead's Process and Reality. In these discussions, the reader will find a lively and learned account of a field of philosophy that is often thought difficult to access, but in this work becomes most accessible and a pleasure to read. -- back cover. (shrink)

Inspired by Cantor's Theorem (CT), orthodoxy takes infinities to come in different sizes. The orthodox view has had enormous influence in mathematics, philosophy, and science. We will defend the contrary view---Countablism---according to which, necessarily, every infinite collection (set or plurality) is countable. We first argue that the potentialist or modal strategy for treating Russell's Paradox, first proposed by Parsons (2000) and developed by Linnebo (2010, 2013) and Linnebo and Shapiro (2019), should also be applied to CT, in a way that (...) vindicates Countabilism. Our discussion dovetails with recent independently developed treatments of CT in Meadows (2015), Pruss (2020), and Scambler (2021), aimed at establishing the mathematical viability, and therefore epistemic possibility, of Countabilism. Unlike these authors, our goal isn't to vindicate the mathematical underpinnings of Countabilism. Rather, we aim to argue that, given that Countabilism is mathematically viable, Countabilism should moreover be regarded as true. After clarifying the modal content of Countabilism, we canvas some of Countabilism's many positive implications, including that Countabilism provides the best account of the pervasive independence phenomena in set theory, and that Countabilism has the power to defuse several persistent puzzles and paradoxes found in physics and metaphysics. We conclude that in light of its theoretical and explanatory advantages, Countabilism is more likely true than not. (shrink)

a. 'Two things fill the mind with ever new and increasing admiration and awe, the oftener and the more steadily we reflect on them: the starry heavens above and the moral law within. ' Thus Kant formulates his attitude to morality (Critique of Practical Reason, p. 260). He draws a sharp distinction between these two objects of admiration. The starry sky, he writes, represents my relationship to the natural, empirical world. Moral law, on the other hand, is of a completely (...) different order. It ' . . . begins from my invisible self, my personality, and exhibits me in a world which has true infinity, but which is traceable only by the understanding and with which I discern that I am not in a merely contingent but in a universal and necessary connection (. . . ). ' (p. 260). So Kant sees morality as a separate metaphysical order opposed to the world of empirical phenomena. Human beings belong to both worlds. According to Kant, the personality derives nothing of value from its relationship with the empirical world. His part in the sensuous world of nature places man on a level with any animal which before long must give back to the rest of nature the substances of which it is made. (shrink)

In this insightful study, Bockja Kim evaluates J.N. Findlay's philosophy of religion in order to determine whether it provides a basis for the positive construction of moral philosophy. In this effort, Kim relies heavily on Hegel's distinction between bad and true infinity to interpret Findlay's philosophical thought. Kim argues that the significance of Findlay's moral philosophy lies in its attempt to construct a method for positive moral reflection by redressing the extreme negative philosophies of transcendentalism and existentialism. Findlay's (...) philosophy thus effectively counters the negativity of moral skepticism and mysticism, both of which tend to neglect moral philosophy and ethics. In addition, Kim explores how Findlay's philosophy may promote inter-faith dialogue between religious traditions of the East and West. Philosophers and religious scholars with interests in Christianity and Buddhism will find this thought-provoking work enlightening. (shrink)

I respond herein to reviews of my recent book by Ann Pederson and Stuart Kurtz. With respect to Pederson's concerns, a constructive theology formulated from the ideas of communication theory need not necessarily neglect pressing historical issues of the poor and powerless. The potential for such relevance remains strong. This is true as well for the application of the system to particular myths and rituals. Also, while I speak positively of computers as instruments of disclosure and the theories upon (...) with they are based as resources for theological construction, this should not be construed as an endorsement of just any application of information technology in a world that tends to distort all good things. With respect to Kurtz's concerns, while thermodynamics plays a role in discussions of the primordial chaos, notions from communication theory are far more central. Also, the use of the language of the theory for theology does not necessarily require theological relevance for all of Claude Shannon's technical conclusions. My uses of infinity are taken from traditional theology and analytic geometry rather than from pure mathematics, although fruitful development along those lines is entirely possible. Pederson and Kurtz are generous with both their praise and concerns. The praise will encourage me to further this project along lines provided by the concerns. (shrink)

I propose a revision of Cantor’s account of set size that understands comparisons of set size fundamentally in terms of surjections rather than injections. This revised account is equivalent to Cantor's account if the Axiom of Choice is true, but its consequences differ from those of Cantor’s if the Axiom of Choice is false. I argue that the revised account is an intuitive generalization of Cantor’s account, blocks paradoxes—most notably, that a set can be partitioned into a set that (...) is bigger than it—that can arise from Cantor’s account if the Axiom of Choice is false, illuminates the debate over whether the Axiom of Choice is true, is a mathematically fruitful alternative to Cantor’s account, and sheds philosophical light on one of the oldest unsolved problems in set theory. (shrink)

Hellman and Shapiro explore the development of the idea of the continuous, from the Aristotelian view that a true continuum cannot be composed of points to the now standard, entirely punctiform frameworks for analysis and geometry. They then investigate the underlying metaphysical issues concerning the nature of space or space-time.