According to Leibniz’s infinite-analysis account of contingency, any derivative truth is contingent if and only if it does not admit of a finite proof. Following a tradition that goes back at least as far as Bertrand Russell, several interpreters have been tempted to explain this biconditional in terms of two other principles: first, that a derivative truth is contingent if and only if it contains infinitely complex concepts and, second, that a derivative truth contains infinitely complex concepts if (...) and only if it does not admit of a finite proof. A consequence of this interpretation is that Leibniz’s infinite-analysis account of contingency falls prey to Robert Adams’s Problem of Lucky Proof. I will argue that this interpretation is mistaken and that, once it is properly understood how the idea of an infinite proof fits into Leibniz’s circle of modal notions, the problem of lucky proof simply disappears. (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)

Leibniz war gewillt, die Idee der kontingenten Wahrheiten über nur mögliche individuelle Substanzen ernst zu nehmen -unabhängig davon, ob diese Substanzen existieren oder nicht. Einer der Wege, diese Idee zu erklären, ist die berühmte Lehre von der unendlichen Analyse. Eine wichtige und verwirrende Schwierigkeit für diese Lehre ist das von Robert M. Adams erörterte Problem des Beweises mit Glück. Auch wenn der vollständige individuelle Begriff einer möglichen Substanz S sich durch Analyse in unendlich viele einfache Begriffe zerlegen läßt, ist es (...) schwerlich erforderlich, daß der Beweis für das Enthaltensein eines Prädikats F im vollständigen Begriff von 5 unendlicher Analyse bedarf: Beweise gehen schrittweise vor; es ist offensichtlich möglich, daß einige Prädikate früher und schneller hergeleitet werden als andere. In diesem Aufsatz zeigen und entwickeln wir zwei leibnizische Wege, die das Problem des mit Glück kurzen Beweises beantworten. (shrink)

Abstract : In this paper, it is argued that Leibniz’s view that necessity is grounded in the availability of a demonstration is incorrect and furthermore, can be shown to be so by using Leibniz’s own examples of infinite analyses. First, I show that modern mathematical logic makes clear that Leibniz’s "infinite analysis" view of contingency is incorrect. It is then argued that Leibniz's own examples of incommensurable lines and convergent series undermine, rather than bolster his view by (...) providing examples of necessary mathematical truths that are not demonstrable. Finally, it is argued that a more modern view on convergent series would, in certain respects, help support some claims he makes about the necessity of mathematical truths, but would still not yield a viable theory of necessity due to remaining problems with other logical, mathematical, and modal claims. (shrink)

I propose a counterfactual theory of infinite regress arguments. Most theories of infinite regress arguments present infinite regresses in terms of indicative conditionals. These theories direct us to seek conditions under which an infinite regress generates an infinite inadmissible set. Since in ordinary language infinite regresses are usually expressed by means of infinite sequences of counterfactuals, it is natural to expect that an analysis of infinite regress arguments should be based on (...) a theory of counterfactuals. The Stalnaker–Lewis theory of counterfactuals, augmented with some fundamental notions from metric-spaces, provides a basis for such an analysis of infinite regress arguments. Since the technique involved in the analysis is easily adaptable to various analyses, it facilitates a rigorous comparison among conflicting philosophical analyses of any given infinite regress. (shrink)

The long-periodic/infinite discrete Gabor transform is more effective than the periodic/finite one in many applications. In this paper, a fast and effective approach is presented to efficiently compute the Gabor analysis window for arbitrary given synthesis window in DGT of long-periodic/infinite sequences, in which the new orthogonality constraint between analysis window and synthesis window in DGT for long-periodic/infinite sequences is derived and proved to be equivalent to the completeness condition of the long-periodic/infinite DGT. By (...) using the property of delta function, the original orthogonality can be expressed as a certain number of linear equation sets in both the critical sampling case and the oversampling case, which can be fast and efficiently calculated by fast discrete Fourier transform. The computational complexity of the proposed approach is analyzed and compared with that of the existing canonical algorithms. The numerical results indicate that the proposed approach is efficient and fast for computing Gabor analysis window in both the critical sampling case and the oversampling case in comparison to existing algorithms. (shrink)

THE MAJOR PREMISE OF DUNS SCOTUS'S IMPRESSIVE PROOF FOR THE EXISTENCE OF GOD HAS BEEN NEGLECTED. THAT PREMISE, "THE MOST PERFECT BEING IS INFINITE," IS ESTABLISHED IN TWO WAYS. THE KEY PREMISE IN EACH WAY IS THE PROPOSITION, "POSSIBLY, SOME BEING IS INFINITE." THIS PROPOSITION CANNOT BE PROVEN TO BE TRUE, NOT BECAUSE IT IS IN ANY WAY DUBIOUS OR LACKING IN EVIDENCE, BUT BECAUSE ITS TERMS ARE SIMPLE AND NOT SUBJECT TO PROOF OR FURTHER ANALYSIS. BEING (...) IS THE SIMPLEST AND MOST IMMEDIATE OF CONCEPTS; AND BEING INFINITE IS SIMPLY THE NEGATION OF BEING FINITE. INSTEAD OF PROOF, SCOTUS PRESENTS "PERSUASIONES", SKETCHES OF LANGUAGE-GAMES TO SHOW THAT THE PROPOSITION IS TRUE. (shrink)

Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time. By doing so, they provide a natural model of infinitary computability, a theoretical setting for the analysis of the power and limitations of supertask algorithms.

Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time. By doing so, they provide a natural model of infinitary computability, a theoretical setting for the analysis of the power and limitations of supertask algorithms.

This article discusses how the concept of a fair finite lottery can best be extended to denumerably infinite lotteries. Techniques and ideas from non-standard analysis are brought to bear on the problem.

There is an infinity of figures and of movements, present and past, which enter into the efficient cause of my present writing, and in its final cause, there are an infinity of slight tendencies and dispositions of my soul, present and past.1.

Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time. By doing so, they provide a natural model of infinitary computability, a theoretical setting for the analysis of the power and limitations of supertask algorithms.

This article presents an analysis of the structure and use of the Finnish independent infinitives. Although typological studies have shown that syntactically independent non-finite constructions are widespread in many languages, the understanding of their semantic and intersubjective motivation is still in its early stages. The current paper aims to enrich the understanding of independent non-finite constructions by closely looking at free-standing infinitive constructions in spoken and written Finnish: it combines theoretical concepts of Cognitive Grammar with the methodological tools of (...) Interactional Linguistics to explore the nature of independent infinitives as a resource for conceptualization and the intersubjective functions that it affords. The paper suggests that the fact that independent infinitives are grammatically ungrounded makes them useful in interactional and textual sequences involving affect display. As the indexical functions of infinitives can be explained from their own morphosyntactic and semantic characteristics, the paper makes the more general claim that there is no synchronic evidence that would support the assumption that such constructions ever evolved, via ellipsis, from finite constructions. Methodologically and theoretically, the paper advocates an approach that takes into account both the social and cognitive nature of language, and promotes the view that Cognitive Grammar offers a flexible, semantically rich starting point for the description of intersubjective meanings conveyed by grammar, when combined with the context-sensitive and microanalytical methodology of Interactional Linguistics. (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)

The concept of an infinite game played on a finite graph is perhaps novel in the context of an rather extensive recent literature in which infinite games are generally played on an infinite game tree. We claim two advantages for our model, which is admittedly more restrictive. First, our games have a more apparent resemblance to ordinary parlor games in spite of their infinite duration. Second, by distinguishing those nodes of the graph that determine the winning (...) and losing of the game , we are able to offer a complexity analysis that is useful in computer science applications. (shrink)

Attempts a faithful account of Bruno's thought as expressed in his writings and to give an analysis of his thought as he developed it in regard to the issue of the infinitive.

Attempts a faithful account of Bruno's thought as expressed in his writings and to give an analysis of his thought as he developed it in regard to the issue of the infinitive.

In this dissertation we discuss infinite regress arguments from both a historical and a logical perspective. Throughout we deal with arguments drawn from various areas of philosophy. ;We first consider the regress generating portion of the argument. We find two main ways in which infinite regresses can be developed. The first generates a regress by defining a relation that holds between objects of some kind. An example of such a regress is the causal regress used in some versions (...) of the Cosmological Argument. The second sort of regress is generated by defining a certain procedure that can be repeated indefinitely many times. This will often be a procedure of analysis or of explanation. ;We then argue that for purposes of discussing regressive viciousness it is helpful to divide regresses into two main types. One type raises primarily epistemological issues. In this class we find the regress of justification often used in foundational arguments in epistemology. We also discuss certain arguments that treat theories of predication as attempts to explain predication. Our other class contains those regresses that raise primarily metaphysical problems. This class includes the causal regress as well as those arguments concerned with predication as an ontological problem. ;We agree with some philosophers that one problem that arises in the case of vicious regression is circularity. We show how this problem can be generalized to include other apparently non-circular regresses. The other main problem in epistemological regresses arises as a result of certain conditions that are placed on the relation involved. We look in detail at the epistemological justification regress to illustrate this last point. ;With respect to metaphysical regresses we again find several types of viciousness arguments. One problem is whether the infinite collection required for the regress is even possible. Another problem is whether an infinite number of events can occur in a finite time. These we deal with only briefly. The main difficulty with these regresses, considered as regresses and not just as infinite sets, again turns out to be certain conditions built into the relation involved. We focus on some arguments about causal regression to illustrate this point. ;Finally, we find no absolutely general conditions for regressive viciousness. There is, however, a certain limited range of ways in which a regress may be vicious. We make these ways explicit. (shrink)

In ‘Fair Infinite Lotteries’ (FIL), Wenmackers and Horsten use non-standard analysis to construct a family of nicely-behaved hyperrational-valued probability measures on sets of natural numbers. Each probability measure in FIL is determined by a free ultrafilter on the natural numbers: distinct free ultrafilters determine distinct probability measures. The authors reply to a worry about a consequent ‘arbitrariness’ by remarking, “A different choice of free ultrafilter produces a different ... probability function with the same standard part but infinitesimal differences.” (...) They illustrate this remark with the example of the sets of odd and even numbers. Depending on the ultrafilter, either each of these sets has probability 1/2, or the set of odd numbers has a probability infinitesimally higher than 1/2 and the set of even numbers infinitesimally lower. The point of the current paper is simply that the amount of indeterminacy is much greater than acknowledged in FIL: there are sets of natural numbers whose probability is far more indeterminate than that of the set of odd or the set of even numbers. (shrink)

Context: The infinite has long been an area of philosophical and mathematical investigation. There are many puzzles and paradoxes that involve the infinite. Problem: The goal of this paper is to answer the question: Which objects are the infinite numbers (when order is taken into account)? Though not currently considered a problem, I believe that it is of primary importance to identify properly the infinite numbers. Method: The main method that I employ is conceptual analysis. (...) In particular, I argue that the infinite numbers should be as much like the finite numbers as possible. Results: Using finite numbers as our guide to the infinite numbers, it follows that infinite numbers are of the structure w + (w* + w) a + w*. This same structure also arises when a large finite number is under investigation. Implications: A first implication of the paper is that infinite numbers may be large finite numbers that have not been investigated fully. A second implication is that there is no number of finite numbers. Third, a number of paradoxes of the infinite are resolved. One change that should occur as a result of these findings is that “infinitely many” should refer to structures of the form w + (w* + w) a + w*; in contrast, there are “indefinitely many” natural numbers. Constructivist content: The constructivist perspective of the paper is a form of strict finitism. (shrink)

Alexander Pruss and I have proposed an analysis of omnipotence which makes no use of the problematic terms 'power' and 'ability'. However, this raises an obvious worry: if our analysis is not related to the notion of power, then how can it count as an analysis of omnipotence, the property of being all-powerful, at all? In this paper, I show how omnipotence can be understood as the possession of infinite power (general, universal, or unlimited power) rather (...) than the possession of particular powers. I then argue that the ordinary notions of particular powers or abilities can be understood as applying vague limitations to the notion of infinite power. The vagueness of these limitations is the source of the well-known difficulties in the analysis of ability. (shrink)

This paper criticizes non-constructive uses of set theory in formal economics. The main focus is on results on preference aggregation and Arrow’s theorem for infinite electorates, but the present analysis would apply as well, e.g., to analogous results in intergenerational social choice. To separate justified and unjustified uses of infinite populations in social choice, I suggest a principle which may be called the Hildenbrand criterion and argue that results based on unrestricted axiom of choice do not meet (...) this criterion. The technically novel part of this paper is a proposal to use a set-theoretic principle known as the axiom of determinacy ), not as a replacement for Choice, but simply to eliminate applications of set theory violating the Hildenbrand criterion. A particularly appealing aspect of \ from the point of view of the research area in question is its game-theoretic character. (shrink)

Postmodern critical assessments of Freud’s theory of mourning disavow the idea of grief’s conclusiveness, insisting that mourning is an interminable process or even a transcendental structure of experience. However, such assessments presuppose an ontological orientation toward finitude that avoids the profound speculative implications of the non-finite status of death in the unconscious. In consequence, mourning comes to assume an indefinite, generic status as a condition of experience instead of a resolutely speculative confrontation with the impossible real of infinitude. Freud’s writings (...) evidence his difficulties with this unconscious disjunction of mourning from finitude, but can be elucidated by turning to Hegel’s critical discussion of “infinite grief,” the melancholy apprehension of finitude as a spurious infinity. Hegel’s analysis in turn anticipates Lacan’s elaboration of the death drive as a concept of unconscious transmission. Here, what begins to emerge is a theory of mourning that concerns the speculative inheritance of an immortal object. (shrink)

We study theorems from Functional Analysis with regard to their relationship with various weak choice principles and prove several results about them: “Every infinite‐dimensional Banach space has a well‐orderable Hamel basis” is equivalent to ; “ can be well‐ordered” implies “no infinite‐dimensional Banach space has a Hamel basis of cardinality ”, thus the latter statement is true in every Fraenkel‐Mostowski model of ; “No infinite‐dimensional Banach space has a Hamel basis of cardinality ” is not provable (...) in ; “No infinite‐dimensional Banach space has a well‐orderable Hamel basis of cardinality ” is provable in ; (the Axiom of Choice for denumerable families of non‐empty finite sets) is equivalent to “no infinite‐dimensional Banach space has a Hamel basis which can be written as a denumerable union of finite sets”; Mazur's Lemma (“If X is an infinite‐dimensional Banach space, Y is a finite‐dimensional vector subspace of X, and, then there is a unit vector such that for all and all scalars α”) is provable in ; “A real normed vector space X is finite‐dimensional if and only if its closed unit ball is compact” is provable in ; (Principle of Dependent Choices) + “ can be well‐ordered” does not imply the Hahn‐Banach Theorem () in ; and “no infinite‐dimensional Banach space has a Hamel basis of cardinality ” are independent from each other in ; “No infinite‐dimensional Banach space can be written as a denumerable union of finite‐dimensional subspaces” lies in strength between (the Axiom of Countable Choice) and ; implies “No infinite‐dimensional Banach space can be written as a denumerable union of closed proper subspaces” which in turn implies ; “Every infinite‐dimensional Banach space has a denumerable linearly independent subset” is a theorem of, but not a theorem of ; and “Every infinite‐dimensional Banach space has a linearly independent subset of cardinality ” implies “every Dedekind‐finite set is finite”. (shrink)

This chapter contains sections titled: Aristotle on the Infinite (to apeiron): From Cosmological Principle to Mathematical Operation Aristotle on Space: Magnitude (megethos) and Place (topos) Aristotle on Time: The “Number of Motion” and “Ever‐rolling Stream” Bibliography.

Traditionally, emancipatory politics is a question of knowing which parts of society are capable of counting for something, and which ones are not. Formulating the question of emancipatory politics in terms of existence, more specifically, in terms of “political subjects who are not social groups but rather forms of inscriptions of the count of the uncounted” , means acknowledging that the proper place for emancipatory politics is the very terrain in which the system of domination operates, a system that radical (...) political theorists characterize as a system of placement, identification, or counting. At present, however, this question of counting the uncountable, crucial for emancipatory politics, cannot be raised at all to the extent that globalization means that everybody is always already included, the exclusion of the uncounted is necessarily obscured, indeed, it has become invisible. Beginning with a discussion on an enigmatic remark: “the infinitization of the value of the subject”, taken from Lacan’s Seminar The Four Fundamental concepts of Psycho-analysis, this paper examines a conceptual shift in the articulation of the relation between the subject and the figures of the Other in an epoch of the Other which does not exist. Taking as her point of departure Lacan’s theory of the cut, the author shows how, in an era of the “generalized metonymization”, only an act constitutes a way out of a discourse that knows no closure. (shrink)

It is possible that a fair coin tossed infinitely many times will always land heads. So the probability of such a sequence of outcomes should, intuitively, be positive, albeit miniscule: 0 probability ought to be reserved for impossible events. And, furthermore, since the tosses are independent and the probability of heads (and tails) on a single toss is half, all sequences are equiprobable. But Williamson has adduced an argument that purports to show that our intuitions notwithstanding, the probability of an (...) infinite sequence is 0. In this paper, I rebut his argument.No Abstract. (shrink)

This paper compares and contrasts two infinite regress arguments against higher-order theories of consciousness that were put forward by the Buddhist epistemologists Dignāga (ca. 480–540 CE) and Dharmakīrti (ca. 600–660). The two arguments differ considerably from each other, and they also differ from the infinite regress argument that scholars usually attribute to Dignāga or his followers. The analysis shows that the two philosophers, in these arguments, work with different assumptions for why an object-cognition must be cognised: for (...) Dignāga it must be cognised in order to enable subsequent memory of it, for Dharmakīrti it must be cognised if it is to cognise an object. (shrink)

With the help of syncretiсs as a new philosophical logic, the philosophy of carriers, the theory of similarity and the theory of Infinite Hierarchical Nesting of Matter, the problems of modern physics are analyzed. We consider the classical and relativistic mechanics, the special and general theories of relativity, the theory of electromagnetic and gravitational fields, of weak and strong interactions. The goal is axiomatization of these theories, building models of elementary particles and of their interactions with each other. The (...) main obtained results are: description of the electrokinetic theory of the origin of magnetic fields in cosmic bodies; calculation of metric in the uniformly accelerated reference frame; the axiomatic construction of electrodynamics, Lorentz invariant (LITG) and covariant theories of gravitation (CTG); comparing CTG with the general theory of relativity and with the results of gravitational experiments. Among other results – the analysis of the properties of ether as the medium responsible for transfer of electromagnetic and gravitational waves, and derivation of the formula for entropy in a tensor form. For students and researchers, as well as for those interested in physical and philosophical problems. (shrink)

By exploiting the parallels between large, yet finite lotteries on the one hand and countably infinite lotteries on the other, we gain insights in the foundations of probability theory as well as in epistemology. We solve the 'adding problems' that occur in these two contexts using a similar strategy, based on non-standard analysis.

Hegel suggests that spirit, in contrast to animal nature, can encounter infinite agony in the death of what was its center, and yet, by dwelling with this loss, emerge into a new form of existence. The paradigm for this move is described toward the end of the chapter on Revealed Religion in the Phenomenology of Spirit. An analysis of the key paragraph introduces a discussion of four questions: Why is this experience triggered by the death of a mediator? (...) What characterizes the spiritual metamorphosis that results? Are such transformations restricted to revealed religion? And what does this defining characteristic tell us about the way spiritual life differs from the natural? (shrink)

In this study, we propose a new direction of research on the axiomatic analysis of approval voting, which is a common democratic decision method. Its novelty is to examine an infinite population setting, which includes an application to intergenerational problems. In particular, we assume that the set of the population is countably infinite. We provide several extensions of the method of approval voting for this setting. As our main result, axiomatic characterizations of the extensions are offered by (...) revealing a direct link between approval voting and the Borda rule. The characterized methods are natural extensions of the standard approval voting method for the finite-population case and are regarded as minimum requirements for other possible infinite-population extensions, which are reasonably democratic. (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)

In his Foundations of a General Theory of Manifolds, Georg Cantor praised Bernard Bolzano as a clear defender of actual infinity who had the courage to work with infinite numbers. At the same time, he sharply criticized the way Bolzano dealt with them. Cantor’s concept was based on the existence of a one-to-one correspondence, while Bolzano insisted on Euclid’s Axiom of the whole being greater than a part. Cantor’s set theory has eventually prevailed, and became a formal basis of (...) contemporary mathematics, while Bolzano’s approach is generally considered a step in the wrong direction. In the present paper, we demonstrate that a fragment of Bolzano’s theory of infinite quantities retaining the part-whole principle can be extended to a consistent mathematical structure. It can be interpreted in several possible ways. We obtain either a linearly ordered ring of finite and infinitely great quantities, or a partially ordered ring containing infinitely small, finite and infinitely great quantities. These structures can be used as a basis of the infinitesimal calculus similarly as in non-standard analysis, whether in its full version employing ultrafilters due to Abraham Robinson, or in the recent “cheap version” avoiding ultrafilters due to Terence Tao. (shrink)

A discussion of the notion of infinity as it appears in the prototype of the great Summae of the thirteenth century, the Summa Fratris Alexandri, traditionally ascribed to Alexander of Hales but now known to be a compilation by the monks at the Paris house of the Franciscans to which Alexander belonged. This Summa reveals the initial effects of the Aristotelian analysis upon the then dominant Neo-Platonic, Augustinian and Anselmian Illuminationisms, and its relation to the philosophical notion of an (...) infinite God. As Wass stresses, the notion of infinity systematically expounded in this Summa provided the conceptual framework for most of the subsequent medieval speculation on the subject.—E. A. R. (shrink)

In 2007. Terence Tao wrote on his blog an essay about soft analysis, hard analysis and the finitization of soft analysis statements into hard analysis statements. One of his main examples was a quasi-finitization of the infinite pigeonhole principle IPP, arriving at the "finitary" infinite pigeonhole principle FIPP₁. That turned out to not be the proper formulation and so we proposed an alternative version FIPP₂. Tao himself formulated yet another version FIPP₃ in a revised (...) version of his essay. We give a counterexample to FIPP₁ and discuss for both of the versions FIPP₂ and FIPP₃ the faithfulness of their respective finitization of IPP by studying the equivalences IPP ↔ FIPP₂ and IPP ↔ FIPP₃ in the context of reverse mathematics ([9]). In the process of doing this we also introduce a continuous uniform boundedness principle CUB as a formalization of Tao's notion of a correspondence principle and study the strength of this principle and various restrictions thereof in terms of reverse mathematics, i.e., in terms of the "big five" subsystems of second order arithmetic. (shrink)

Catherine Malabou, Antonio Negri, John D. Caputo, Bruno Bosteels, Mark C. Taylor, and Slavoj Zizek join seven others--including William Desmond, Katrin Pahl, Adrian Johnston, Edith Wyschogrod, and Thomas A. Lewis--to apply Hegel's thought to twenty-first-century philosophy, politics, and religion. Doing away with claims that the evolution of thought and history is at an end, these thinkers safeguard Hegel's innovations against irrelevance and, importantly, reset the distinction of secular and sacred. These original contributions focus on Hegelian analysis and the transformative (...) value of the philosopher's thought in relation to our current "turn to religion." Malabou develops Hegel's motif of confession in relation to forgiveness; Negri writes of Hegel's philosophy of right; Caputo reaffirms the radical theology made possible by Hegel; and Bosteels critiques fashionable readings of the philosopher and argues against the reducibility of his dialectic. Taylor reclaims Hegel's absolute as a process of infinite restlessness, and Zizek revisits the religious implications of Hegel's concept of letting go. Mirroring the philosopher's own trajectory, these essays progress dialectically through politics, theology, art, literature, philosophy, and science, traversing cutting-edge theoretical discourse and illuminating the ways in which Hegel inhabits them. (shrink)

Hume on Infinite Divisibility and Sensible Extensionless Indivisibles DALE JACQUETTE 'Twere certainly to be wish'd, that some expedient were fallen upon to reconcile philosophy and common sense, which with regard to the question of infinite divisibility have wag'd most cruel wars with each other. David Hume, A Treatise of Human Nature 1. THE DIVISIBILITY ARGUMENTS David Hume's refutation of the infinite divisibility of space and time, and his doctrine of the sensible extensionless indivisibles that constitute extension, are (...) perhaps the least loved, and until recently, least examined aspects of his early philosophy. The fact that Hume reduced his lengthy discussion of the prob- lem from the Treatise to just two pages of the first Enquiry, together with his disavowal of the first work in the Enquiry's "Advertisement," have reinforced the tendency to ignore his arguments. ~ Opinions on the merits of Hume's analysis of space are widely divided. To give a sense of the disparity, Donald L. M. Baxter, in "Hume on Infinite Divisibility," maintains: "Far from begging the question, Hume has available a respectable argument against the infinite divisibility of finite spatial intervals."' C. D. Broad, on the contrary, in his Dawes Hicks Lecture on Philosophy to the British Academy, "Hume's Doctrine of Space," concludes: "I think, then, that Hume's whole account of spatial divisibility can David Hume, Essays and Treatises on... (shrink)

Until the Scientific Revolution, the nature and motions of heavenly objects were mysterious and unpredictable. The Scientific Revolution was revolutionary in part because it saw the advent of many mathematical tools—chief among them the calculus—that natural philosophers could use to explain and predict these cosmic motions. Michel Blay traces the origins of this mathematization of the world, from Galileo to Newton and Laplace, and considers the profound philosophical consequences of submitting the infinite to rational analysis. "One of Michael (...) Blay's many fine achievements in _Reasoning with the Infinite_ is to make us realize how velocity, and later instantaneous velocity, came to play a vital part in the development of a rigorous mathematical science of motion."—Margaret Wertheim, _New Scientist_. (shrink)

Until the Scientific Revolution, the nature and motions of heavenly objects were mysterious and unpredictable. The Scientific Revolution was revolutionary in part because it saw the advent of many mathematical tools—chief among them the calculus—that natural philosophers could use to explain and predict these cosmic motions. Michel Blay traces the origins of this mathematization of the world, from Galileo to Newton and Laplace, and considers the profound philosophical consequences of submitting the infinite to rational analysis. "One of Michael (...) Blay's many fine achievements in _Reasoning with the Infinite_ is to make us realize how velocity, and later instantaneous velocity, came to play a vital part in the development of a rigorous mathematical science of motion."—Margaret Wertheim, _New Scientist_. (shrink)