On the standard interpretation, Aristotle's conception of "nous" is geared against skeptical worries about the possibility of scientific knowledge and ultimately of the knowledge of first principles. On this view, Aristotle introduces "nous" as an intuitive faculty that grasps the first principles once and for all as true in such a way that it does not leave any room for the skeptic to press his skeptical point any further. This position views Aristotelian "nous" as having an internalist justificatory role in (...) Aristotelian epistemology. In contrast, a minority (empiricist) position holds the same internalist justificatory view of "nous" but rejects its internally certifiable infallibility by stressing the connection between "nous" and Aristotelian induction. I argue that both approaches are flawed in that Aristotle's project in the "Posterior Analytics" is not to answer the skeptic on internalist justificatory grounds, but rather lay out a largely externalist "explication" of scientific knowledge. (shrink)

In this paper I formulate the thesis of the Division of Epistemic Labor as a thesis of epistemic dependence, illustrate several ways in which individual subjects are epistemically dependent on one or more of the members of their community in the process of knowledge acquisition, and draw conclusions about the cognitively distributed nature of some knowledge acquisition.

This paper presents a simple mathematical model that shows how economic inequality between social groups can arise and be maintained even when the only adaptive learning process driving cultural evolution increases individuals’ economic gains. The key assumptions are that human populations are structured into groups and that cultural learning is more likely to occur within than between groups. Then, if groups are sufficiently isolated and there are potential gains from specialization and exchange, stable stratification can sometimes result. This model predicts (...) that stratification is favored, ceteris paribus, by (1) greater surplus production, (2) more equitable divisions of the surplus among specialists, (3) greater cultural isolation among subpopulations within a society, and (4) more weight given to economic success by cultural learners. (shrink)

This chapter focuses on one of the common fallacies in Western philosophy called 'division'. The fallacy of division occurs when one incorrectly infers that the characteristics, attributes, or features of the group as a whole will also be found in the individuals comprising the group. The easiest way to avoid this fallacy is never to assume that the characteristics, attributes, or features of the group as a whole will also be found in the individuals comprising the group. One (...) must inspect and evaluate the characteristics, attributes, or features of the whole separately from the parts of which the whole is comprised. Inferences from a whole to a part can be made, however, if additional assumptions are added to the argument that link together the property had by the whole with the property had by the part. (shrink)

Aquinas accepts that (i) some kinds of voluntary action are (qua voluntary) “basic,” not divisible into (non-fictional) further kinds; (ii) a concrete individual action may belong to more than one basic kind; (iii) the basic kinds to which it belongs are determined by the agent’s intentions qua performing the action; (iv) some intentions may stand to others as means to ends; (v) there can be concrete individual actions in which the agent’s intended means are disordered with respect to the ends; (...) (vi) such actions are morally wrong; (vii) whether a given intention is disordered as means to a given end is determined solely by the natures of the agent and of the intended means or ends. Together, these propositions entail that, pace many analytic philosophers, concrete individual actions can have a moral wrongness that consists neither in expectation of disutility nor in violation of the pure logic of practical reason. (shrink)

The paper mobilizes Adam Smith’s treatment of the division of labour in relation to the production, consumption and exchange of knowledge. One aspect of this mobilization deals with the epistemic demands that exchange makes on its participants. The other deals with increasing returns in the provision of knowledge itself, treating knowledge creation as just another example of specialization and exchange. These two aspects come together in relation to the epistemic demands associated with assessing knowledge quality. These demands differ according (...) to whether the knowledge is embodied in products or whether the knowledge is an object for its own sake. It is argued that disciplines play a critical role as institutions for meeting the epistemic demands that the division of labour creates in the ‘knowledge’ case. (shrink)

In this article we revisit a number of disputes regarding significance logics---i.e., inferential frameworks capable of handling meaningless, although grammatical, sentences---that took place in a series of articles most of which appeared in the Australasian Journal of Philosophy between 1966 and 1978. These debates concern (i) the way in which logical consequence ought to be approached in the context of a significance logic, and (ii) the way in which the logical vocabulary has to be modified (either by restricting some notions, (...) or by adding some vocabulary) to keep as much of Classical Logic as possible. Our aim is to show that the divisions arising from these disputes can be dissolved in the context of a novel and intuitive proposal that we put forward. (shrink)

Milner and Goodale's influential account of the primate cortical visual streams involves a division of consciousness between them, for it is the ventral stream that has the responsibility for visual consciousness. Hence, the dorsal visual stream is a ‘zombie’ stream. In this article, I argue that certain information carried by the dorsal stream likely plays a central role in the egocentric spatial content of experience, especially the experience of visual spatial constancy. Thus, the dorsal stream contributes to a pervasive (...) feature of consciousness. (shrink)

In the first book of the Prior Analytics, Aristotle sets out, for the first time in Greek philosophy, a logical system. It consists of a deductive system (I.4-22), meta-logical results (I.23-26), and a method for finding and giving deductions (I.27-29) that can apply in “any art or science whatsoever” (I.30). After this, Aristotle compares this method with Plato’s method of division, a procedure designed to find essences of natural kinds through systematic classification. This critical comparison in APr I.31 raises (...) an interpretive puzzle: how can Aristotle reasonably juxtapose two methods that differ so much in their aims and approach? What can be gained by doing so? Previous interpreters have failed to show how this comparison is legitimate or what important point Aristotle is making. The goal of this paper is to resolve the puzzle. In resolving this puzzle we not only learn more about the relationship be- tween division and the syllogistic in Aristotle. We will also learn something about the motivation for the syllogistic itself, by seeing the role that it plays in his philosophy of science. (shrink)

Recent debates over adults' theory of mind use have been fueled by surprising failures of perspective-taking in communication, suggesting that perspective-taking may be relatively effortful. Yet adults routinely engage in effortful processes when needed. How, then, should speakers and listeners allocate their resources to achieve successful communication? We begin with the observation that the shared goal of communication induces a natural division of labor: The resources one agent chooses to allocate toward perspective-taking should depend on their expectations about the (...) other's allocation. We formalize this idea in a resource-rational model augmenting recent probabilistic weighting accounts with a mechanism for (costly) control over the degree of perspective-taking. In a series of simulations, we first derive an intermediate degree of perspective weighting as an optimal trade-off between expected costs and benefits of perspective-taking. We then present two behavioral experiments testing novel predictions of our model. In Experiment 1, we manipulated the presence or absence of occlusions in a director–matcher task. We found that speakers spontaneously modulated the informativeness of their descriptions to account for “known unknowns” in their partner's private view, reflecting a higher degree of speaker perspective-taking than previously acknowledged. In Experiment 2, we then compared the scripted utterances used by confederates in prior work with those produced in interactions with unscripted directors. We found that confederates were systematically less informative than listeners would initially expect given the presence of occlusions, but listeners used violations to adaptively make fewer errors over time. Taken together, our work suggests that people are not simply “mindblind”; they use contextually appropriate expectations to navigate the division of labor with their partner. We discuss how a resource-rational framework may provide a more deeply explanatory foundation for understanding flexible perspective-taking under processing constraints. (shrink)

In this article, I will begin by describing recent divisive concepts legislation, which bans teaching about aspects of racism, sexism, and equity, speculating briefly on the motivations behind it and the implications resulting from it. I will then describe how discussing divisive concepts in classrooms may be a helpful way for students to better understand the particular concepts and for students to take a stand on them. While I will briefly argue for the importance of classroom discussion of divisive concepts, (...) my central claim will be that we must do more than merely discuss these concepts. Instead, we should engage in classroom inquiry about them. Inquiry positions students to not be silent regarding divisive concepts, as some legislation may cause, and inquiry positions students to not just talk about divisive concepts, but to solve social problems related to them in our world today. In other words, inquiry enables us to do something about divisive concepts, rather than just learning about them, as some educators currently advocate in calls for discussion. And inquiry is certainly more effective than banning divisive concepts or pushing them to the background, as recent legislation attempts to do. I argue that rather than excluding divisive concepts or the divisions they may provoke from our classrooms, we need educative spaces where we inquire into what divides us, why we are divided, and how we might respond to such division. Within that inquiry, we can better attend to our divisions while also taking up the specific concepts prohibited in recent legislation. (shrink)

In Plato’s Statesman, the Eleatic Stranger applies a specialized method of inquiry—the “method of collection and division”, or “method of division”—in order to discover the nature of statecraft. This paper articulates some consequences of the fact that the method is both a tool for identifying natural kinds—that is, a tool for carving the world by its joints (Phaedrus 265b-d)—and social kinds—that is, the kinds depending on human beings for their existence and explanation. A central goal of the paper (...) is to illuminate the extent to which this use of the method of division allows us to identify Plato as an early historical forerunner of racialism, which is an ideology according to which humanity divides into races differentiated by heritable physiological, cultural, and intellectual traits, as a way of vindicating oppressive and exploitative social, political, and economic systems. -/- I defend an interpretation of the Stranger’s claim, much discussed in the literature, that the division of humankind into Greek and barbarian is unnatural (Politicus 262c-263a). I argue that, in the Stranger’s view, this division reflects subjective illusion and prejudice, rather than the fundamental, and teleological, structure of human social organization, which concerns how human beings rationally cooperate to self-produce as a species. Nonetheless, the Stranger’s alternative theory of the natural structure of human society, I suggest, is proto-racial in another way. Through a brief consideration of the Stranger’s affirmative and complex division of kinds in the city, I argue that he re-introduces naturalistic foundations for unjust human hierarchies through his alternative theory of natural kinds and human social teleology. (shrink)

Cutting a cake, dividing up the property in an estate, determining the borders in an international dispute - such problems of fair division are ubiquitous. Fair Division treats all these problems and many more through a rigorous analysis of a variety of procedures for allocating goods, or deciding who wins on what issues, when there are disputes. Starting with an analysis of the well-known cake-cutting procedure, 'I cut, you choose', the authors show how it has been adapted in (...) a number of fields and then analyze fair-division procedures applicable to situations in which there are more than two parties, or there is more than one good to be divided. In particular they focus on procedures which provide 'envy-free' allocations, in which everybody thinks he or she has received the largest portion and hence does not envy anybody else. They also discuss the fairness of different auction and election procedures. (shrink)

We further investigate a divisibility relation on the set of BN ultrafilters on the set of natural numbers. We single out prime ultrafilters (divisible only by 1 and themselves) and establish a hierarchy in which a position of every ultrafilter depends on the set of prime ultrafilters it is divisible by. We also construct ultrafilters with many immediate successors in this hierarchy and find positions of products of ultrafilters.

According to a standard interpretation of Hume’s argument against infinite divisibility, Hume is raising a purely formal problem for mathematical constructions of infinite divisibility, divorced from all thought of the stuffing or filling of actual physical continua. I resist this. Hume’s argument must be understood in the context of a popular early modern account of the metaphysical status of the parts of physical quantities. This interpretation disarms the standard mathematical objections to Hume’s reasoning; I also defend it on textual and (...) contextual grounds. (shrink)

Some hold that the lesson of Russell’s paradox and its relatives is that mathematical reality does not form a ‘definite totality’ but rather is ‘indefinitely extensible’. There can always be more sets than there ever are. I argue that certain contact puzzles are analogous to Russell’s paradox this way: they similarly motivate a vision of physical reality as iteratively generated. In this picture, the divisions of the continuum into smaller parts are ‘potential’ rather than ‘actual’. Besides the intrinsic interest of (...) this metaphysical picture, it has important consequences for the debate over absolute generality. It is often thought that ‘indefinite extensibility’ arguments at best make trouble for mathematical platonists; but the contact arguments show that nominalists face the same kind of difficulty, if they recognize even the metaphysical possibility of the picture I sketch. (shrink)

The question of the division of cognitive labor has given rise to various models characterizing the way scientists should distribute their efforts. These models often consider the scientific community as a self-governed sphere constituted by rational agents making choices on the basis of fixed rules. Such models have recently been criticized for not taking into account the real mechanisms of science funding. Hence, the question of the utility of the DCL models in guiding science policy remains an open one. (...) In this paper, we show that two unconsidered dimensions would have to be taken into account. First, DCL studies miss the existence of distinct levels of epistemic objectives organizing the research process. Indeed, the scientific field is structured as a system of hierarchical, interconnected practices which are defined both by their inherent purposes and by various superposed external functions. Second, I criticize the absence of ontological considerations, since the epistemological significance of pluralism is highly dependent on the nature of the object under study. Because of these missing dimensions, current DCL models might have a limited usefulness to identify good practices of research governance. (shrink)

Morality, according to some theories, demands a lot of us. One way to defend such demanding moral theories is through an appeal to the division of normativity; on this picture, morality is only one of the normative domains that guides us, so it should be expected that we often fail to follow that guidance. This paper defends the division of normativity as a response to demandingness objections against an alternative: moral rationalism. It does this by addressing and refuting (...) three arguments: the argument from blameworthiness, the argument from agency, and the argument from authority. In turn, I show that none of these arguments work as responses to the division of normativity – if normativity generally is divided, so too must be blameworthiness, agency, and authority. (shrink)

Justice makes demands upon us. But these demands, important though they may be, are not the only moral demands that we face. Our lives ought to be responsive to other values too. However, some philosophers have identified an apparent tension between those values and norms, such as justice, that seem to transcend the arena of small-scale interpersonal relations and those that are most at home in precisely that arena. How, then, are we to engage with all of the values and (...) norms that we take to apply to us? In this article, I discuss one way that we might hope to resolve the tension and its relation to John Rawls's `basic structure restriction'. The prospect of resolution is offered by the idea of a `division of moral labour', according to which the pursuit of certain values is assigned to institutions and not to individuals. According to Rawls's basic structure restriction, principles of justice are applicable only to the institutions of the basic structure of society. The possibility of a connection between the division of moral labour and the basic structure restriction readily suggests itself. Taking G.A. Cohen's well-known `incentives' critique of the basic structure restriction as a starting point, I consider five ways in which that restriction might be defended by appeal to the division of moral labour. I conclude that none of these defences succeeds, for none convinces that the conditions in which it makes sense to apply the division of moral labour idea obtain for Rawls's conception of distributive justice. Although the division of moral labour is an attractive proposal, it can do no work in a Rawlsian context. Key Words: Cohen • distributive justice • egalitarian ethos • equality • Rawls. (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)

How do scientists generate knowledge in groups, and how have they done so in the past? How do epistemically motivated social interactions influence or even drive this process? These questions speak to core interests of both history and philosophy of science. Idealised models and formal arguments have been suggested to illuminate the social epistemology of science, but their conclusions are not directly applicable to scientific practice. This paper uses one of these models as a lens and historiographical tool in the (...) examination of actual scientific collectives. It centres on the analysis of two episodes from the history of photosynthesis research of the late nineteenth- to mid-twentieth centuries, which display a wide and coordinated intellectual diversity similar to Kitcher’s “division of cognitive labour”. The concept, I argue, captures important aspects of the photosynthesis research communities, but the underlying process unfolded in ways that differ from the model’s assumption in interesting ways. The paper unravels how the self-organised interplay of cooperation and competition, and the dynamics of individual and collective goals within scientific communities were influential factors in the generation of knowledge. From there, some thoughts are developed on how historical and philosophical approaches in the analysis of science can productively interact. (shrink)

In this paper, we address an infamous argument against divisibility that dates back to Zeno. There has been an incredible amount of discussion on how to understand the critical notions of divisibility, extension, and infinite divisibility that are crucial for the very formulation of the argument. The paper provides new and rigorous definitions of those notions using the formal theories of parthood and location. Also, it provides a new solution to the paradox of divisibility which does not face some threats (...) that can possibly undermine the standard Lebesgue measure solution to such a paradox. (shrink)

The UK-based deliberations that led up to the legalisation of two new ‘mitochondrial donation’ techniques in 2015, and which continued after that time as regulatory details were determined, featured a division of advisory labour that is common when decisions are made about new technologies. An expert panel was convened by the Human Fertilisation and Embryology Authority, charged with assessing the scientific and technical aspects of these techniques. Meanwhile, the Nuffield Council on Bioethics addressed the ethical issues. While this (...) class='Hi'>division of labour was undertaken in the name of thoroughness, I argue here that it can have the unintended consequence that hybrid questions that simultaneously involve ethical and technical aspects—especially questions about where to set evidential thresholds for the acceptance of new technology—do not receive enough attention. (shrink)

The UK-based deliberations that led up to the legalisation of two new ‘mitochondrial donation’ techniques in 2015, and which continued after that time as regulatory details were determined, featured a division of advisory labour that is common when decisions are made about new technologies. An expert panel was convened by the Human Fertilisation and Embryology Authority (HFEA), charged with assessing the scientific and technical aspects of these techniques. Meanwhile, the Nuffield Council on Bioethics addressed the ethical issues. While this (...) division of labour was undertaken in the name of thoroughness, I argue here that it can have the unintended consequence that hybrid questions that simultaneously involve ethical and technical aspects—especially questions about where to set evidential thresholds for the acceptance of new technology—do not receive enough attention. (shrink)

[ Samuel Scheffler] Some egalitarian liberals have proposed a division of moral labour between social institutions and individual agents, but the division-of-labour metaphor has been understood in different ways. This paper aims to disentangle some of these different understandings, with an eye to clarifying the appeal of the egalitarian-liberal project and the challenges that it faces. The idea of a division of moral labour is best understood as the expression of a strategy for accommodating diverse values. It (...) is not an apology for economic self-interest or a device for justifying personal acquisitiveness. /// [Véronique Munoz-Dardé] Are there distinctively political values? Certain egalitarians seem to think that equality is one such value. Scheffler's contribution to the symposium seeks to articulate a division of moral labour between norms of personal morality and the principles of justice that regulate social institutions, and using this suggests that the egalitarian critique of Rawls can be deflected. In this paper, instead, I question the status of equality as an intrinsic value. I argue that an egalitarianism which focuses on the status of equality as valuable in itself embraces a theory of value with the worst elements of utilitarianism while leaving behind any of the intuitive appeal that utilitarianism has. In its place I press that we need a political conception of egalitarianism which stresses the role we engage beyond those found in the norms of personal morality. (shrink)

The question of the division of cognitive labor has given rise to various models characterizing the way scientists should distribute their efforts. These models often consider the scientific community as a self-governed sphere constituted by rational agents making choices on the basis of fixed rules. Such models have recently been criticized for not taking into account the real mechanisms of science funding. Hence, the question of the utility of the DCL models in guiding science policy remains an open one. (...) In this paper, we show that two unconsidered dimensions would have to be taken into account. First, DCL studies miss the existence of distinct levels of epistemic objectives organizing the research process. Indeed, the scientific field is structured as a system of hierarchical, interconnected practices which are defined both by their inherent purposes and by various superposed external functions. Second, I criticize the absence of ontological considerations, since the epistemological significance of pluralism is highly dependent on the nature of the object under study. Because of these missing dimensions, current DCL models might have a limited usefulness to identify good practices of research governance. (shrink)

Aristotle's syllogistic theory, as developed in his Prior Analytics, is often regarded as the birth of logic in Western philosophy. Over the past century, scholars have tried to identify important precursors to this theory. I argue that Platonic division, a method which aims to give accounts of essences of natural kinds by progressively narrowing down from a genus, influenced Aristotle's logical theory in a number of crucial respects. To see exactly how, I analyze the method of division as (...) it was originally conceived by Plato and received by Aristotle. I argue that, while Plato allowed that some divisions fail to rigorously investigate the essence, he began a program continued by Aristotle (and others in antiquity and the middle ages) of seeking norms for division that would apply in any domain whatsoever. This idea of a rigorous, general method was taken up and developed by Aristotle in his syllogistic. Aristotle also used Plato's conception of predication as parthood in his semantics for syllogistic propositions. As part of my argument, I prove that a semantics based on Platonic divisional structures is sound and complete for the deduction system used in the literature to model Aristotle's syllogistic. (shrink)

The notion of the ‘division of physiological labour’ is today an outdated relic in the history of science. This contrasts with the fate of another notion, which was so frequently paired with the division of physiological labour, which is the concept of ‘morphological differentiation.’ This is one of the elementary modal concepts of ontogenesis. In this paper, we intend to target the problems and causes that gradually led biologists to combine these two notions during the 19th century, and (...) to progressively dissociate them, retaining only the concept of differentiation by the early 20th century. We shall adhere to the following: 1. The primitive economic concept of the division of labour is not a descriptive notion denoting a type of organisation of labour, but an etiological one: the idea of a causal relationship between this type of organization and the improvement of the whole. 2. This concept rapidly interested naturalists such as Henri Milne-Edwards, who were keen to find a rational ground for hierarchizing living forms based on anatomical complexity. 3. The validation of this notion in the realms of biology was subject to at least two conditions which were far from being fully satisfied. This did not prevent, however, the initial success of the concept of the division of physiological labour during the second half of the 19th century. 4. Finally, the gradual disqualification, within the Darwinian theoretical context, of the conception of an intrinsic hierarchical rank of organisms, led to a lack of interest in the concept of the physiological division of labour, at least in its non-Darwinian and non-ecological variant. (shrink)

Psychologists have used experimental methods to study language for more than a century. However, only with the recent availability of large-scale linguistic databases has a more complete picture begun to emerge of how language is actually used, and what information is available as input to language acquisition. Analyses of such “big data” have resulted in reappraisals of key assumptions about the nature of language. As an example, we focus on corpus-based research that has shed new light on the arbitrariness of (...) the sign: the longstanding assumption that the relationship between the sound of a word and its meaning is arbitrary. The results reveal a systematic relationship between the sound of a word and its meaning, which is stronger for early acquired words. Moreover, the analyses further uncover a systematic relationship between words and their lexical categories—nouns and verbs sound differently from each other—affecting how we learn new words and use them in sentences. Together, these results point to a division of labor between arbitrariness and systematicity in sound-meaning mappings. We conclude by arguing in favor of including “big data” analyses into the language scientist's methodological toolbox. (shrink)

It is shown that a stable division ring with positive characteristic has finite dimension over its centre. This is then extended to simple division rings.

In Republic II Glaucon assigns to Socrates the task of praising justice for itself. What it means to praise justice for itself is unclear. A new interpretation is offered on the basis of an analysis of Glaucon's division of goods. A distinction is developed between criterial benefits, those valuable consequences of a thing which provide a standard for evaluating a thing as a good instance of its type, and fringe benefits, valuable consequences which do not provide such a standard. (...) Socrates is expected to praise justice by describing the benefits it constitutes as a valuable activity of soul. He may also use the criterial benefits of justice but not its fringe benefits. This account of Socrates' task is superior to those interpretations which rule out all use of valuable consequences in praising justice and to those interpretations which fail to preserve the distinction between the second and third classes of goods. (shrink)

It seems plausible that (i) how much punishment a person deserves cannot be affected by the mere existence or nonexistence of another person. We might have also thought that (ii) how much punishment is deserved cannot increase merely in virtue of personal division. I argue that (i) and (ii) are inconsistent with the popular belief that, other things being equal, when people culpably do very wrong or bad acts, they ought to be punished for this—even if they have repented, (...) are now virtuous, and punishing them would benefit no one. Insofar as we cannot deny (i), we are either forced to abandon the popular belief in desert, or else allow that personal division could, as I put it, “multiply desert.” Some may not find the latter, considered by itself, troubling. But I argue that the thesis that division multiplies desert faces a potentially serious problem, which arises in the context of personal fusion. It is difficult to see how to maintain a particular family of desert views in light of the cases here presented. (shrink)

The phenomenon of twinning in early fetal development has become a popular source for doubt regarding the ascription of moral status to early embryos. In this paper, the possible moral basis for such a line of reasoning is critically analysed with sceptical results. Three different versions of the argument from twinning are considered, all of which are found to rest on confusions between the actual division of embryos involed in twinning and the property of early embryos to be divisible, (...) be based on highly questionable ethical assumptions, or to imply inconsistent claims regarding the moral importance of potentiality and/or the moral status of embryos. This is taken to expose a number of related inconsistencies in the moral basis of pro-life positions. In particular, ascribing moral significance to the property of being (in)divisible is found to be incompatible with the claim that human individuals possess unique values which could underpin an absolute moral ban on murder. (shrink)

We introduceself-divisibleultrafilters, which we prove to be precisely those$w$such that the weak congruence relation$\equiv _w$introduced by Šobot is an equivalence relation on$\beta {\mathbb Z}$. We provide several examples and additional characterisations; notably we show that$w$is self-divisible if and only if$\equiv _w$coincides with the strong congruence relation$\mathrel {\equiv ^{\mathrm {s}}_{w}}$, if and only if the quotient$(\beta {\mathbb Z},\oplus )/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}$is a profinite group. We also construct an ultrafilter$w$such that$\equiv _w$fails to be symmetric, and describe the interaction between the (...) aforementioned quotient and the profinite completion$\hat {{\mathbb Z}}$of the integers. (shrink)

Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the ‘three-fold way’. (...) It is perhaps easiest to see it in the study of irreducible unitary representations of groups on complex Hilbert spaces. These representations come in three kinds: those that are not isomorphic to their own dual (the truly ‘complex’ representations), those that are self-dual thanks to a symmetric bilinear pairing (which are ‘real’, in that they are the complexifications of representations on real Hilbert spaces), and those that are self-dual thanks to an antisymmetric bilinear pairing (which are ‘quaternionic’, in that they are the underlying complex representations of representations on quaternionic Hilbert spaces). This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics. More generally, Hilbert spaces of any one of the three kinds—real, complex and quaternionic—can be seen as Hilbert spaces of the other kinds, equipped with extra structure. (shrink)

Aristotle proposed in the famous part of his 6th Book of Metaphysics a triple categorization of speculative sciences (physics, mathematics, first philosophy). Some scholars have considered this classical division as an unlawful and inconsistent Platonic residue of the parallelism between knowledge and reality. Others have numbered Aristotle among the upholders of the scholastic doctrine of the so-called degrees of abstraction. After having profoundly examined texts of the Corpus Aristotelicum (Metaphysics E 1, 1025b 1 - 1026a 32; On the soul (...) A 1, 403a 27 - 403b 19) not only we can refute the accusation of Platonism and the unlawful and anachronistic attribution of degrees of abstraction doctrine to the Stagirite, but principally understand the real importance of Aristotle's proposition. That is to say: its role in determining the epistemic character of the science on which all this deep inquiry takes place: the "filosofia prima". (shrink)

Increasingly, both mothers and fathers are expected to play an equal role in child rearing. Nonetheless, we know little about how child care arrangements affect couples’ sexual intimacy and relationship quality. Research has focused on the effect of the division of paid labor and housework on couples’ relationships, finding that egalitarianism is problematic for sexual intimacy, relationship quality, and relationship stability. These findings, however, come almost universally from studies utilizing decades-old data that fail to examine the division of (...) child care. In this study, we update this work by utilizing data from the 2006 Marital and Relationship Study to examine how the division of child care affects the relationship quality and sexual intimacy of heterosexual couples in the United States. Results indicate that men’s performance of child care is generally associated with more satisfaction with the division of child care, more satisfying sexual relationships, and higher quality relationships. Importantly, we find that egalitarian child care arrangements have positive consequences for both men and women. These findings contribute to a growing body of research that challenges the costs of egalitarianism and indicates instead that egalitarianism is associated with higher quality, more intimate relationships than gender traditional arrangements. (shrink)

In the Statesman (287c3-5), Plato proposes that the philosophical divider should divide analogously to how the butcher divides a sacrificial animal. According to the common interpretation, the example of animal sacrifice illustrates that we should “cut off limbs” (kata mele), that is, divide non-dichotomously into functional parts of a living whole. We argue that this interpretation is historically inaccurate and philosophically problematic: it relies on an inaccurate understanding of sacrificial butchery and leads to textual puzzles. Against the common interpretation, we (...) argue that the example of animal sacrifice illustrates that correct division minimizes (it cuts into the smallest number possible) by first dividing dichotomously and then dividing non-dichotomously into “parts,” not “limbs.” We will show that both the philosophical divider and sacrificial butcher proceed exactly in this way. By taking Plato’s comparison to the historical practice of animal sacrifice seriously, our interpretation provides better solutions to the textual puzzles than the common interpretation. (shrink)