We report the results of a questionnaire study on the fair distribution of indivisible goods. We collected data from three different subject pools, first- and second- year students majoring in economics, law students, and advanced economics students with some background knowledge of fairness theories. The purpose of this study is to assess the empirical relevance of various fairness criteria such as inequality aversion, the utilitarian principle of maximizing the sum of individual payoffs, the Rawlsian “maximin” principle of maximizing the payoff (...) of the worst-off individual, and the criterion of envy-freeness (in the sense of Foley 1967). (shrink)

This paper provides a short historical and systematic survey of parameters, problems, and proposals concerning the theoretical treatment of indivisible temporal boundaries throughout the ages. A very early trace of thinking about them is identified in Aristophanes’ comedy The Clouds. The approach of logicians in the late Middle Ages is placed in a broad context. Links of this topic to the issues of vagueness, modality, space and quantized time are discussed.

Physical boundaries and the earliest topologists. Topology has a relatively short history; but its 19th century roots are embedded in philosophical problems about the nature of extended substances and their boundaries which go back to Zeno and Aristotle. Although it seems that there have always been philosophers interested in these matters, questions about the boundaries of three-dimensional objects were closest to center stage during the later medieval and modern periods. Are the boundaries of an object actually existing, less-than-three-dimensional parts of (...) the object—that is, are solids bounded by two-dimensional surfaces, surfaces by one-dimensional “edges” or “physical lines”, edges by dimensionless “simples”? If not, how does a perfectly spherical object manage to touch a perfectly flat object—what part of the sphere is in immediate contact with the plane, if the sphere has no unextended parts? But if such parts be admitted, are we not then saddled with “actual infinities” of simples, lines, and surfaces spread throughout each continuous object—the boundaries of all the object’s internal parts? Does it help to say that these internal boundaries exist only “potentially”? (shrink)

The interpretation of quantum mechanics presented in this paper is inspired by two ideas that are fundamental in Bohr’s writings: indivisibility and complementarity. Further basic assumptions of the proposed interpretation are completeness, universality and conceptual economy. In the interpretation, decoherence plays a fundamental role for the understanding of measurement. A general and precise conception of complementarity is proposed. It is fundamental in this interpretation to make a distinction between ontological reality, constituted by everything that does not depend at all on (...) the collectivity of human beings, nor on their decisions or limitations, nor on their existence, and empirical reality constituted by everything that not being ontological is, however, intersubjective. According to the proposed interpretation, neither the dynamical properties, nor the constitutive properties of microsystems like mass, charge and spin, are ontological. The properties of macroscopic systems and space-time are also considered to belong to empirical reality. The acceptance of the above mentioned conclusion does not imply a total rejection of the notion of ontological reality. In the paper, utilizing the Aristotelian ideas of general cause and potentiality, a relation between ontological reality and empirical reality is proposed. Some glimpses of ontological reality, in the form of what can be said about it, are finally presented. (shrink)

Physical boundaries and the earliest topologists. Topology has a relatively short history; but its 19th century roots are embedded in philosophical problems about the nature of extended substances and their boundaries which go back to Zeno and Aristotle. Although it seems that there have always been philosophers interested in these matters, questions about the boundaries of three-dimensional objects were closest to center stage during the later medieval and modern periods. Are the boundaries of an object actually existing, less-than-three-dimensional parts of (...) the object—that is, are solids bounded by two-dimensional surfaces, surfaces by one-dimensional “edges” or “physical lines”, edges by dimensionless “simples”? If not, how does a perfectly spherical object manage to touch a perfectly flat object—what part of the sphere is in immediate contact with the plane, if the sphere has no unextended parts? But if such parts be admitted, are we not then saddled with “actual infinities” of simples, lines, and surfaces spread throughout each continuous object—the boundaries of all the object’s internal parts? Does it help to say that these internal boundaries exist only “potentially”? (shrink)

The main purpose of this article is to present new material concerning Leibniz's use of indivisibles and infinitesimals in his early mathematical texts. Most of these texts are contained in hitherto unpublished manuscripts and are soon to be printed in volume VII, 4 of the Academy Edition. They present examples which illustrate how Leibniz operated with concepts such as indivisibles and infinitesimals in that period of his development.

The Invisible Remainder' begins with a detailed examination of the two works in which Schelling's speculative audacity reached its peak: his essay on human freedom and his drafts on the 'Ages of the World.

This article defends a novel, normative conception of the indivisibility of human rights. Human rights are indivisible because normative commitment to one mutually entails normative commitment to another. The normative conception enables us to defend three important theoretical and practical corollaries. First, as a conceptual thesis normative indivisibility lets us see how human rights constitute a unified system not liable to the typical counter-examples to indivisibility as mutual indispensability. Second, as a dialectical thesis, normative indivisibility can support linkage arguments in (...) defense of controversial human rights. And third, as a political thesis, normative indivisibility can show why the political thesis of indivisibility means that states lack discretion to ‘pick and choose’ which human rights to implement. (shrink)

Hegel’s idealism is generally perceived as a system of rational sublation (Aufhebung) of all empirical contingencies: nothing resists notional mediation which, in a movement of negation of negation, establishes a rational totality. Already Schelling opposed to this complete sublation an “indivisible remainder” of empirical contingency. However, a close reading of Hegel makes it clear that the concluding moment of a dialectical movement of sublation is an empirical remainder which totalizes it, like the body of Christ in Christianity. And the same (...) goes for the process of “negation of negation”: it concludes with a failure of negation, and its ultimate form is a failed suicide where the subject survives as a living dead. Such a reading of Hegel makes him a thinker of our time which is the time of a failed negation: most of us live as survivors of our death, with life dragging on in depressive apathy. (shrink)

Every set can been thought of as a directed graph whose edge relation is ∈. We show that many natural examples of directed graphs of this kind are indivisible: for every infinite κ, for every indecomposable λ, and every countable model of set theory. All of the countable digraphs we consider are orientations of the countable random graph. In this way we find indivisible well‐founded orientations of the random graph that are distinct up to isomorphism, and ℵ1 that are distinct (...) up to siblinghood. (shrink)

In this paper I reconstruct and discuss Antonio Rubio (1546-1615)’s theory of the composition of the continuum, as set out in his Tractatus de compositione continui, a part of his influential commentary on Aristotle’s Physics, published in 1605 but rewritten in 1606. Here I attempt especially to show that Rubio’s is a significant case of Scholastic overlapping between Aristotle’s theory of infinitely divisible parts and indivisibilism or ‘Zenonism’, i.e. the theory that allows for indivisibles, extensionless points, lines, and surfaces, (...) which are supposed to take part in the composition of the continuum. Even if such a syncretic tendency was, in many different ways, already developing in the medieval period and then at the end of the sixteenth century, Rubio’s position is indeed peculiar. He maintains that indivisibles are real and actual, infinite in act, really distinct from each other, and that, although they indwell in substance, indivisibles do not contribute directly to the constitution of the continuum. In this reconstruction I emphasize notably Rubio’s usage of mereological notions like those of part, whole, completeness and incompleteness. (shrink)

As its modern edition appears in the Synthese Historical Library, Adam WodehamThis book is an important contribution to the history of philosophy.It will be of interest to all medievalists, particularly to those concerned with medieval science, philosophy, and logic. Theologians and historians of mathematics will also find it useful.Whether charity or [any] other incorruptible form is composed of indivisible forms.Because this difficulty is the same for all composite divisible things, whether intensive or extensive, which are of one and the same (...) species or homogeneous, therefore I will briefly inquire indifferently concerning the former and the latter.” 3 The solutions Wodeham then proposes to the questions he asks rely nearly always on logic. (shrink)

In “The Jesuits and the Method of Indivisibles” David Sherry criticizes a central thesis of my book Infinitesimal: that in the seventeenth century the Jesuits sought to suppress the method of indivisibles because it undermined their efforts to establish a perfect rational and hierarchical order in the world, modeled on Euclidean Geometry. Sherry accepts that the Jesuits did indeed suppress the method, but offers two objections. First, that the book does not distinguish between indivisibles and infinitesimals, and (...) that whereas the Jesuits might have reason to object to the first, the second posed no problem for them. Second, seeking an alternative explanation for the Jesuits’ hostility to the method, he proposes that its implicit atomism conflicted with the Catholic doctrine of the sacrament of the Eucharist, and was therefore heretical. In response to Sherry’s first criticism I point out that the Jesuits objected to all forms of the method of indivisibles, and that the distinction between indivisibles and infinitesimals consequently cannot explain the fight over the method in the seventeenth century. With regards to the Eucharist, I agree with Sherry that the Jesuits were indeed concerned about the method’s affinity to atomism and materialism, though for a different reason: these doctrines were antithetical to their efforts to impose divine hierarchy and order on the world. In as much as the technical details of the miracle of the Eucharist mattered, they provided no grounds for objecting to a mathematical doctrine. (shrink)

The name of Democritus can claim a place in any discussion of indivisibles. Yet its introduction in this paper seems to depend on the lucus a non lucendo principle; for Democritus did not believe in the existence of indivisible lines. Nowhere is the belief ascribed to him and in at least one place it is implicitly denied, the scholion on De Caelo 268a 1, which says he made his elements indivisible solids, as contrasted with lines or surfaces. Two passages, (...) one from Plutarch, the other from Simplicius, will show why he could not believe in indivisible lines. (shrink)

As its modern edition appears in the Synthese Historical Library, Adam Wodeham’s Tractatus de indivisibilibus does not appear to belong to any one discipline. With regard to its intended audience, the notice of the book appearing on the back cover states that “This book is an important contribution to the history of philosophy.” But it continues, “It will be of interest to all medievalists, particularly to those concerned with medieval science, philosophy, and logic. Theologians and historians of mathematics will also (...) find it useful.”Adam de Wodeham, Tractatus de indivisibilibus. A Critical Edition with Introduction, Translation, and Textual Notes, ed. Rega Wood (Dordrecht: Kluwer Academic Publishers, 1988). In its medieval context as well, Tractatus de indivisibilibus had ambiguous disciplinary status. It begins with the question, “Whether charity or [any] other incorruptible form is composed of indivisible forms.”Wodeham, Tractatus de indivisibilibus, p. 33. Such a reference to charity signals a connection to the Sentences of Peter Lombard, Book I, dist. 17. In introducing his answer to this question, however, Wodeham states, “Because this difficulty is the same for all composite divisible things, whether intensive or extensive, which are of one and the same species or homogeneous, therefore I will briefly inquire indifferently concerning the former and the latter.”Wodeham, Tractatus de indivisibilibus, pp. 34–35. Here and elsewhere, I occasionally modify Rega Wood’s translation in the interests of exactness. The solutions Wodeham then proposes to the questions he asks rely nearly always on logic. (shrink)

As its modern edition appears in the Synthese Historical Library, Adam Wodeham’s Tractatus de indivisibilibus does not appear to belong to any one discipline. With regard to its intended audience, the notice of the book appearing on the back cover states that “This book is an important contribution to the history of philosophy.” But it continues, “It will be of interest to all medievalists, particularly to those concerned with medieval science, philosophy, and logic. Theologians and historians of mathematics will also (...) find it useful.”Adam de Wodeham, Tractatus de indivisibilibus. A Critical Edition with Introduction, Translation, and Textual Notes, ed. Rega Wood (Dordrecht: Kluwer Academic Publishers, 1988). In its medieval context as well, Tractatus de indivisibilibus had ambiguous disciplinary status. It begins with the question, “Whether charity or [any] other incorruptible form is composed of indivisible forms.”Wodeham, Tractatus de indivisibilibus, p. 33. Such a reference to charity signals a connection to the Sentences of Peter Lombard, Book I, dist. 17. In introducing his answer to this question, however, Wodeham states, “Because this difficulty is the same for all composite divisible things, whether intensive or extensive, which are of one and the same species or homogeneous, therefore I will briefly inquire indifferently concerning the former and the latter.”Wodeham, Tractatus de indivisibilibus, pp. 34–35. Here and elsewhere, I occasionally modify Rega Wood’s translation in the interests of exactness. The solutions Wodeham then proposes to the questions he asks rely nearly always on logic. (shrink)

Philosophical theories of fairness propose to divide a good that several individuals have a claim to in proportion to the strength of their respective claims. We suggest that currently, these theories face a dilemma when dealing with a good that is indivisible. On the one hand, theories of fairness that use weighted lotteries are either of limited applicability or fall prey to an objection by Brad Hooker. On the other hand, accounts that do without weighted lotteries fall prey to three (...) fairness paradoxes. We demonstrate that division methods from apportionment theory, which has hitherto been ignored by philosophical theories of fairness, can be used to provide fair division for indivisible goods without weighted lotteries and without fairness paradoxes. (shrink)

This paper analyzes criteria of fair division of a set of indivisible items among people whose revealed preferences are limited to rankings of the items and for whom no side payments are allowed. The criteria include refinements of Pareto optimality and envy-freeness as well as dominance-freeness, evenness of shares, and two criteria based on equally-spaced surrogate utilities, referred to as maxsum and equimax. Maxsum maximizes a measure of aggregate utility or welfare, whereas equimax lexicographically maximizes persons' utilities from smallest to (...) largest. The paper analyzes conflicts among the criteria along with possibilities and pitfalls of achieving fair division in a variety of circumstances. (shrink)

Pour Aristote, sous le rapport de sa composition en parties, le continu est divisible mais sous le rapport de ses limites (point, ligne, surface et profondeur), le continu est indivisible. Walter Burley, comme ses contemporains, a commenté la coexistence problématique de la divisibilité et de l’indivisibilité dans la structure du continu. Bien plus, aux prises avec sa célèbre polémique contre son adversaire Guillaume d’Ockham à propos de l’ontologie de la catégorie de quantité, il admet une structure du continu originale qui (...) semble contenir à la fois des intervalles ou parties divisibles et des points ou indivisibles. (shrink)

The article begins with the redefinition of complexity and risk. Indeed, phenomena such as earthquakes, pandemics, ecological emergencies, and issues related to the development of technology highlight the unique and reciprocal relationship between complexity and risk. However, modernity endeavoured to simplify complexity and to erase the connection of the latter with any issue concerning risk. Despite its negative results, whose ineffectiveness and dangerousness have at the present become unmistakably clear, the attitude in favour of simplification succeeded in becoming the forma (...) mentis of modern science, politics, culture, ethics, etc. Yet, in the last decades a new trend seems to have arisen, namely the one focusing on the “governance” of complexity and of the related risks. If considered under a socio-political point of view, its aim is to succeed in efficiency, whilst maintaining democracy. This can be achieved through the advancement of dialogue, the appreciation of diversities, and the enhancement of pluralism. Hence, the pars construens of the article focuses on the notion of responsibility, and tries to highlight its fruitfulness for the socio-political “governance” of complexity. (shrink)

Pour Aristote, sous le rapport de sa composition en parties, le continu est divisible mais sous le rapport de ses limites (point, ligne, surface et profondeur), le continu est indivisible. Walter Burley, comme ses contemporains, a commenté la coexistence problématique de la divisibilité et de l’indivisibilité dans la structure du continu. Bien plus, aux prises avec sa célèbre polémique contre son adversaire Guillaume d’Ockham à propos de l’ontologie de la catégorie de quantité, il admet une structure du continu originale qui (...) semble contenir à la fois des intervalles ou parties divisibles et des points ou indivisibles. (shrink)

ABSTRACT Morafe ke bongwe bjo bosa kgaoganego eupja setlwaedi se laetja kgaogano go ya ka merafe Dinyakishisho di supa gore magareng ga batho, morafe ke yo tee fela; ke morafe wa batho. Ya go bitjwa DNA ka Sekgowa e laetja go sena pelaelo gore batho kamoka ke bana ba legoro le lelapa le tee. Ka bjalo, morafe wa batho ga o a tshwanela go kgaolwa dikgaokgao. Bophelong bja ka metlha re bona gore ba gona bao ba gananago le taba ye. (...) Ba ema ka la gore go nale merafe ye mentshi. Go ya ka bona, merafe ye e a fapafapana ebile tlhago e laetja gore go nale merafe ya kua godimo le ya kua fase. Go nale merafe yewe e tlhabologilego le ye esa tlhabologang. Kgopolo ye e tloga ele ditjie badimo. Ke yona e tlholago dintwa. E ganana le phedisano yeo e ithekgilego godimo ga toka le khutjo magareng ga batho. Ka fao taodiso ye e shireletja maemo a gore go nale morafe yo tee fela; morafe yo ebego lelapa le tee fela la batho kamoka. [Race is an indivisible singular but practice insists it is a frangible plural Biological anthropology suggests that there is only one human race. Studies of the DNA have confirmed that there is only one human family marked by a variety of differences such as the type of hair, the color of eyes and skin. Instead of accepting the scientific finding that the human race is one, humanity is involved in an absurd but deadly race to prove that there are many races arising from a preestablished ontological hierarchy. According to this imaginary hierarchy, races are graded on a scale of “superiority” and “inferiority,” “civilization” and “barbarism.” The thesis defended in this article is that the prevailing practice of fragmenting the human race into multiple bits and pieces is against the pursuit of justice and peace in human relations. The human race is an indivisible singular and not a frangible plural.]. (shrink)

This paper defends the claim that the contemporary canon of human rights forms an indivisible and interdependent system of norms against both "Western" and "Asian" critics who have asserted exceptionalist or selectivist counterclaims. After providing a formal definition of human rights, I argue that the set of particular human rights that comprises the contemporary canon represents an ethical-legal paradigm which functions as an implicit theory of human oppression. On this view, human rights originate as normative responses to particular historical experiences (...) of oppression. Since historically known experiences of oppression have resulted from practices that function as parts of systems of domination, normative responses to these practices have sought to disarm and dismantle such systems by depriving potential oppressors of the techniques which enable them to maintain their domination. Therefore, human rights norms form a systematic and interdependent whole because only as parts of a system can they function as effective means for combatting oppression and domination. (shrink)

The pascalian use of indivisibles is here considered in the context of the theological and mathematical debates of the time, by distinguishing it clearly from this of Cavalieri. The combinatory and geometrical approaches are closely linked in Pascal’s work. His use of indivisibles has a heuristic, inventive character and not only a demonstrative one. Ontologically speaking, it stems out from the acceptance of actual infinite. The use of the symmetry axiom of Archimedes is the basis of the pascalian (...) use of the infinitesimals, which has, in other respects, some close connexions with the Leibnizian conception of infinitesimals. (shrink)

A community faces the obligation of providing an indivisible public good that each of its members is able to provide at a certain cost. The solution is to rely on the member who can provide the public good at the lowest cost, with a due compensation from the other members. This problem has been studied in a non-cooperative setting by Kleindorfer and Sertel. They propose an auction mechanism that results in an interval of possible individual contributions whose lower bound is (...) the equal division. Here, instead we take a cooperative stand point by modelling this problem as a cost sharing game that turns out to be a ‘reverse’ airport game whose core is shown to have a regular structure. This enables an easy calculation of the nucleolus that happens to define the upper bound of the Kleindorfer–Sertel interval. The Shapley value instead is not an appropriate solution in this context because it may imply compensations to non-providers. (shrink)