The Schur-Zassenhaus Theorem is one of the fundamental theorems of finite group theory. Here is its statement:Fact1.1 (Schur-Zassenhaus Theorem). Let G be a finite group and let N be a normal subgroup of G. Assume that the order ∣N∣ is relatively prime to the index [G:N]. Then N has a complement in G and any two complements of N are conjugate in G.The proof can be found in most standard books in group theory, e.g., in [S, Chapter 2, Theorem 8.10]. (...) The original statement stipulated one ofNorG/Nto be solvable. Since then, the Feit-Thompson theorem [FT] has been proved and it forces eitherNorG/Nto be solvable. (The analogous Feit-Thompson theorem for groups of finite Morley rank is a long standing open problem).The literal translation of the Schur-Zassenhaus theorem to the finite Morley rank context would state that in a groupGof finite Morley rank a normal π-Hall subgroup (if it exists at all) has a complement and all the complements are conjugate to each other. (Recall that a groupHis called aπ-group, where π is a set of prime numbers, if elements ofHhave finite orders whose prime divisors are from π. Maximal π-subgroups of a groupGare called π-Hall subgroups. They exist by Zorn's lemma. Since a normal π-subgroup ofGis in all the π-Hall subgroups, if a group has a normal π-Hall subgroup then this subgroup is unique.)The second assertion of the Schur-Zassenhaus theorem about the conjugacy of complements is false in general. As a counterexample, consider the multiplicative group ℂ* of the complex number field ℂ and consider thep-Sylow for any primep, or even the torsion part of ℂ*. LetHbe this subgroup.Hhas a complement, but this complement is found by Zorn's Lemma (consider a maximal subgroup that intersectsHtrivially) and the use of Zorn's Lemma is essential. In fact, by Zorn's Lemma, any subgroup that has a trivial intersection withHcan be extended to a complement ofH. Since ℂ* is abelian, these complements cannot be conjugated to each other. (shrink)

One of the purposes of this paper is to prove a partial Schur-Zassenhaus Theorem for groups of finite Morley rank.Theorem 2.Let G be a solvable group of finite Morley rank. Let π be a set of primes, and let H ⊲ G a normal π-Hall subgroup. Then H has a complement in G.This result has been proved in [1] with the additional assumption thatGis connected, and thought to be generalized in [2] by the authors of the present article. Unfortunately in (...) the last section of the latter paper there is an irrepairable mistake. Here we give a new proof of the Schur-Zassenhaus Theorem using the results of [2] up to the last section and a new result that we are going to state below.The second author has shown in [11] that a nilpotentω-stable group is the central product of a divisible subgroup and a subgroup of bounded exponent, generalizing a well-known result of Angus Macintyre about abelian groups [8]. One could ask a similar question for solvable groups: are they a product of two subgroups, one divisible, one of bounded exponent? One is allowed to be hopeful because of the well-known decomposition of the connected solvable algebraic groups over algebraically closed fields as the product of the unipotent radical and a torus. (shrink)

Cauchy’s contribution to the foundations of analysis is often viewed through the lens of developments that occurred some decades later, namely the formalisation of analysis on the basis of the epsilon-delta doctrine in the context of an Archimedean continuum. What does one see if one refrains from viewing Cauchy as if he had read Weierstrass already? One sees, with Felix Klein, a parallel thread for the development of analysis, in the context of an infinitesimal-enriched continuum. One sees, with Emile Borel, (...) the seeds of the theory of rates of growth of functions as developed by Paul du Bois-Reymond. One sees, with E. G. Björling, an infinitesimal definition of the criterion of uniform convergence. Cauchy’s foundational stance is hereby reconsidered. (shrink)

We classify irreducible actions of connected groups of finite Morley rank on abelian groups of Morley rank 3.

A construction of the real number system based on almost homomorphisms of the integers $\mathbb {Z}$ was proposed by Schanuel, Arthan, and others. We combine such a construction with the ultrapower or limit ultrapower construction to construct the hyperreals out of integers. In fact, any hyperreal field, whose universe is a set, can be obtained by such a one-step construction directly out of integers. Even the maximal (i.e., On -saturated) hyperreal number system described by Kanovei and Reeken (2004) and independently (...) by Ehrlich (2012) can be obtained in this fashion, albeit not in NBG . In NBG , it can be obtained via a one-step construction by means of a definable ultrapower (modulo a suitable definable class ultrafilter). (shrink)

We examine Paul Halmos’ comments on category theory, Dedekind cuts, devil worship, logic, and Robinson’s infinitesimals. Halmos’ scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is “certainty” and “architecture” yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to (...) teach us the opposite lesson, namely that the castle is floating in midair. Halmos’ realism tends to color his judgment of purely scientific aspects of logic and the way it is practiced and applied. He often expressed distaste for nonstandard models, and made a sustained effort to eliminate first-order logic, the logicians’ concept of interpretation, and the syntactic vs semantic distinction. He felt that these were vague, and sought to replace them all by his polyadic algebra. Halmos claimed that Robinson’s framework is “unnecessary” but Henson and Keisler argue that Robinson’s framework allows one to dig deeper into set-theoretic resources than is common in Archimedean mathematics. This can potentially prove theorems not accessible by standard methods, undermining Halmos’ criticisms. (shrink)

Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy’s proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy’s proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy’s proof closely and show that it finds closer proxies in a different modern framework.

O objetivo deste artigo é propor, de modo simples e saudável, uma releitura transformadora na relação entre os ministérios ordenados e a missão leiga, sobretudo no tocante à espiritualidade laical como colaboração eclesial. Com efeito, desde o tempo de Jesus, o protagonismo leigo, como cooperador à evangelização sempre foi ressaltado e muito bem aproveitado pelo Mestre; dessa forma, nota-se o quanto a Igreja carece hoje dessa ajuda primordial. Não por menos, o Concilio Vaticano II, no decreto Apostolicam Actuositatem, diz que (...) o fiel leigo é indispensável á missão da Igreja. Assim, entre tantas orientações, o mesmo decreto convoca a renovar e valorizar o ardor missionário com o apostolado leigo no interior da Igreja. Nesse sentido, é urgente a aplicação do pedido solicitado no Concílio Vaticano II para os tempos atuais, visando à sua concretização e não ficar somente em uma longínqua teoria. É nesta óptica, que a V CELAM, realizada em Aparecida, enfatiza mais uma vez a indispensabilidade do leigo como colaborador profícuo na missão eclesial. Por fim, vemos em muito a realização exemplar de muitos fiéis leigos batizados em seu exercício de ministros não-ordenados que oferecem subsídios riquíssimos para santificar a vida na Igreja. (shrink)

TOPOGRAPHICAL MISSIONS ΤΟ LATO I (2005-2007): A CRITICAL ANALYSIS OF THE PLAN BY J. DEMARGNE AND V. SEYK (1901) In the years 2005 and 2007 the site of Lato in Crete was the object of three new topographical and geomorphological study campaigns. The works carried out over the entire site permitted in particular the analysis of the only to date available complete plan of the city, that of J. Demargne and the architect V. Seyk undertaken in 1900. The objectif was (...) to establish its documentary interest and limits in use, and thus better prepare new measurements and guide renewed field work. To do this, it was first necessary (I) to retrace the history of previous graphic documentation, (II) to analyse the work of J. Demargne and V. Seyk, so as (III) to gauge its value against the standards of current research, and (IV) to establish its interest in terms of the cultural resource management of the site. A second paper to appear in BCH will develop, on basis of a new documentation, the urban and morphological analysis of the remains of Lato. (shrink)

COSTA, Alexandre da. Fundamentos da bioética: estudo sobre o pensamento de Hugo Tristram Engelhardt Jr. FERREIRA, Amauri Carlos. O imaginário religioso e modos de vida urbana: experiência e memória da Juventude Católica em Belo Horizonte – MG, anos 80. GONÇALVES, Davidson Sepini. O Panóptico de Jeremy Bentham: por uma leitura utilitarista. GÓIS, Aurino José. Parque Municipal de Belo Horizonte: público, apropriações e significados. FARIA, Paulo Antônio Couto. Teologia no limiar da filosofia: a modernidade e o encontro fé e razão (...) na obra de Henrique Cláudio de Lima Vaz. (shrink)

La polémique entre V. Soloviev et L. Lopatine met en lumière les recherches de la philosophie russe sur son mode propre de compréhension du Moi. Pour Soloviev, on ne peut attribuer au sujet de la conscience en tant que tel une réalité autre que phénoménologique. Lopatine propose une variante, proche de Leibniz, selon laquelle le Moi est une conscience subjective, présente aux différents niveaux du développement spirituel. Cette interprétation est reprise et développée, dans une certaine mesure, par N. Losski. G. (...) Chpet tire un bilan de la discussion, en posant le caractère « communautaire » [ obščnost ’] particulier du Moi, qui se réalise non par la « généralisation » [ obobščenie ] mais par la « communication » [ obščenie ]. (shrink)

This is a long-awaited reprint of the major work of the late Alexandre Koyré, one of the greatest masterpieces of the history of Western philosophy. The early nineteenth century saw a revival of Boehme studies but even the commentaries of Franz von Baader could not be substituted for a scholarly synthesis of Boehme's thought. Boehme is a universe of his own, and only the immense learning of a man like Koyré, with his rare ability to synthesize and clarify, could (...) have done justice to him. Through and beyond the rich and baroque splendor of the cosmic poetry of the Dawn of the Mysterium Magnum, we see the birth of the modern philosophical spirit. What Boehme offers us is a philosophy of freedom, and, despite the modesty of his comparisons, Koyré does not fail to show how all this anticipates the greatest Idealistic systems.--M. J. V. (shrink)

What proper role should considerations of risk, particularly to research subjects, play when it comes to conducting research on human enhancement in the military context? We introduce the currently visible military enhancement techniques (1) and the standard discussion of risk for these (2), in particular what we refer to as the ‘Assumption’, which states that the demands for risk-avoidance are higher for enhancement than for therapy. We challenge the Assumption through the introduction of three categories of enhancements (3): therapeutic, preventive, (...) and pure enhancements. This demands a revision of the Assumption (4), alongside which we propose some further general principles bearing on how to balance risks and benefits in the context of military enhancement research. We identify a particular type of therapeutic enhancements as providing a more responsible path to human trials of the relevant interventions than pure enhancement applications. Finally, we discuss some possible objections to our line of thought (5). While acknowledging their potential insights, we ultimately find them to be unpersuasive, at least provided that our proposal is understood as fully non-coercive towards the candidates for such therapeutic enhancement trials. (shrink)

According to a common philosophical intuition, the deep nature of things is hidden from us, and the world as we know it through perception and science is, just like a dream, shadows, or a computer simulation, somehow shallow and lacking in reality. This “intuition of unreality” clashes with a strong, but perhaps more naive, intuition to the effect that the world as we know it seems perfectly real. Shadows, dreams, or informational structures appear too unreal to be identical to the (...) world as we know it! This clash between the two intuitions forms the basis of the “problem of reality.” In the late nineteenth century psychiatrists encountered patients they referred to as “metaphysician doubters” who constantly questioned the reality of the world. This essay draws on studies of these patients in order to reject, and indeed diagnose, the intuition of unreality and recent metaphysical doctrines drawing on it, such as structuralism, digitalism, and virtual realism. (shrink)

This collection of six essays centers on Professor Koyre's great theme: the relative importance of metaphysics and observation, with controlled experiment a kind of marriage between the two. Professor Koyre's thesis might be summed up as a claim that when one is seeking to explain the scientific revolution, attention must be concentrated on the philosophical outlook of the scientist and away from speculative theories. At the time of his death, Alexandre Koyre was a professor at the Ecole Pratique des (...) Hautes Etudes (Sorbonne) and a memeber of the Institute for Advanced Study in Princeton. (shrink)

In a previous article on cosmological arguments, I have put forward a few examples of complete infinite and circular explanations, and argued that complete non-wellfounded explanations such as these might explain the present state of the world better than their well-founded theistic counterparts (Billon, 2021). Although my aim was broader, the examples I gave there implied merely causal explanations. In this article, I would like to do three things: • Specify some general informative conditions for complete and incomplete non-wellfounded causal (...) explanations that can be used to assess candidate explanations and to generate new examples of complete non-wellfounded explanations. • Show that these conditions, which concern chains of causal explanations, easily generalize to chains of metaphysical, grounding explanations and even to chains involving other “determination relations” such as supervenience. • Apply these general conditions to the recent debates against the existence of nonwellfounded chains of grounds and show, with a couple of precise examples, that the latter can be complete, and that just like in the case of causal explanations, non-wellfoundedness can in fact be an aset rather than a liability. (shrink)

Computational reproducibility possesses its own dynamics and narratives of crisis. Alongside the difficulties of computing as an ubiquitous yet complex scientific activity, computational reproducibility suffers from a naive expectancy of total reproducibility and a moral imperative to embrace the principles of free software as a non-negotiable epistemic virtue. We argue that the epistemic issues at stake in actual practices of computational reproducibility are best unveiled by focusing on software as a pivotal concept, one that is surprisingly often overlooked in accounts (...) of reproducibility issues. Software is not only about designing and coding but also about maintaining, supporting, distributing, licensing, and governance; it is not only about developers but also about users. We focus on openness debates among computational chemists involved in molecular modeling software packages as empirical grounding for our argument. We then identify and analyse four epistemic characteristics as key to the role of software in computational reproducibility. (shrink)

. Computational chemistry grew in a new era of “desktop modeling,” which coincided with a growing demand for modeling software, especially from the pharmaceutical industry. Parameterization of models in computational chemistry is an arduous enterprise, and we argue that this activity leads, in this specific context, to tensions among scientists regarding the epistemic opacity transparency of parameterized methods and the software implementing them. We relate one flame war from the Computational Chemistry mailing List in order to assess in detail the (...) relationships between modeling methods, parameterization, software and the various forms of their enclosure or disclosure. Our claim is that parameterization issues are an important and often neglected source of epistemic opacity and that this opacity is entangled in methods and software alike. Models and software must be addressed together to understand the epistemological tensions at stake. (shrink)

Patients suffering from depersonalization complain of feeling detached from their body, their mental states, and actions or even from themselves. In this chapter, I argue that depersonalization consists in the lack of a phenomenal feature that marks my experiences as mine, which is usually called “mineness,” and that the study of depersonalization constitutes a neglected yet incomparable probe to assess empirically the scope, role, and even the nature of mineness. Here is how I will proceed. After describing depersonalization (§2) and (...) arguing that it involves a lack of mineness (§3) I will confront a series of objections (§4). I will then spell out what depersonalization can teach us about the scope of mineness (§5), about its role and psychological function (§6) and even about its nature (§7). (shrink)

Using emotional film clips is one of the most popular and effective methods of emotion elicitation. The main goal of the present study was to develop and test the effectiveness of a new and comprehensive set of emotional film excerpts. Fifty film experts were asked to remember specific film scenes that elicited fear, anger, sadness, disgust, amusement, tenderness, as well as emotionally neutral scenes. For each emotion, the 10 most frequently mentioned scenes were selected and cut into film clips. Next, (...) 364 participants viewed the film clips in individual laboratory sessions and rated each film on multiple dimensions. Results showed that the film clips were effective with regard to several criteria such as emotional discreteness, arousal, positive and negative affect. Finally, ranking scores were computed for 24 classification criteria: Subjective arousal, positive and negative affect (derived from the PANAS; Watson & Tellegen, 1988), a positive and a negative affect scores derived from the Differential Emotions Scale (DES; Izard et al., 1974), six emotional discreteness scores (for anger, disgust, sadness, fear, amusement and tenderness), and 15 “mixed feelings” scores assessing the effectiveness of each film excerpt to produce blends of specific emotions. In addition, a number of emotionally neutral film clips were also validated. The database and editing instructions to construct the film clips have been made freely available in a website. (shrink)

According to what we will call subjectivity theories of consciousness, there is a constitutive connection between phenomenal consciousness and subjectivity: there is something it is like for a subject to have mental state M only if M is characterized by a certain mine-ness or for-me-ness. Such theories appear to face certain psychopathological counterexamples: patients appear to report conscious experiences that lack this subjective element. A subsidiary goal of this chapter is to articulate with greater precision both subjectivity theories and the (...) psychopathological challenge they face. The chapter’s central goal is to present two new approaches to defending subjectivity theories in the face of this challenge. What distinguishes these two approaches is that they go to great lengths to interpret patients’ reports at face value – greater length, at any rate, than more widespread approaches in the extant literature. (shrink)

This paper focuses on one particular case that connects climate justice and climate economics. Its contribution is twofold. First, it aims at providing a sound normative foundation for carbon pricing mechanisms around the notions of a ‘right to energy’, the ‘duty not-to-harm’ and an argument for ‘restricted compensation’. Second, it identifies the normative elements from theories of climate justice that should guide the design of market-based instruments for climate change mitigation. This will cast light on the particular moral relevance of (...) the act of internalizing a negative externality and of the funds generated by putting a price on carbon. (shrink)

Depersonalization is a pathological condition consisting in a deep modification of the way things appear to a subject, leading him to feel estranged from his body, his actions, his thoughts, his mind and even from himself. In this article, I argue that the study of depersonalization raises three challenges for recent theories of the sense of bodily ownership. These challenges—which I call the centrality challenge, the dissociation challenge and the grounding challenge— thwart most of these theories and suggest that the (...) sense of bodily ownership hinges on a phenomenal mark of mineness that can not be accounted for in terms of our sensory, interoceptive, agentive, cognitive or affective dispositions and that is psychologically primitive. In short: that mineness is first. (shrink)

Depersonalization consists in a deep modification of the way things appear to a subject, leading him to feel estranged from his body, his actions, his thoughts, and his mind, and even from himself. Even though, when it was discovered at the end of the 19th century, this psychiatric condition was widely used to probe certain aspects of bodily awareness, and more specifically the sense of bodily ownership (SBO), it has been strangely neglected in contemporary debates. In this chapter, I argue (...) that because of three specific features, depersonalization raises some important challenges for current theories of the SBO. The first feature — call it “generality” — is that depersonalization does not only affect the sense of bodily ownership but also, typically, the sense of ``mental ownership’’ (SMO), the sense of agency or ``action ownership’’ (SOA), and the subject’s core sense of herself (CSS), that is, her awareness of herself as an I. The second feature is that except for the symptoms of depersonalization, depersonalized patients are hard to distinguish, psychologically, from normal subjects. This makes it hard to find psychological features that might explain their condition. The last feature, call it “fundamentality” is that the psychological features that do seem abnormal among depersonalized patients seem more likely to be explained by depersonalization than to explain it. These three features raise three challenges — the centrality challenge, the dissociation challenge, and the grounding challenge. Taken together, I will argue, these challenges suggest that the SBO depends on a form of phenomenal “mineness” that would mark my mental states as mine and that cannot be accounted for in sensorimotor, cognitive, or even affective terms. A phenomenal mineness that indeed seems to be psychologically primitive, and only accountable in neurophysiological terms. (shrink)

Natural modalities are often analysed from an abstract point of view where they are associated with putative laws of nature. However, the way possibilities are represented in physics is more complex. Lagrangian mechanics, for instance, involves two different layers of modalities: kinematical and dynamical possibilities. This paper examines the status of these two layers, both in the classical and quantum case. The quantum case is particularly problematic: we identify four possible interpretive options. The upshot is that a close inspection of (...) the way possibilities are represented in physics could lead to new ways of thinking about natural modalities. (shrink)

Several advocates of the lively field of “metaphysics of science” have recently argued that a naturalistic metaphysics should be based solely on current science, and that it should replace more traditional, intuition-based, forms of metaphysics. The aim of the present paper is to assess that claim by examining the relations between metaphysics of science and general metaphysics. We show that the current metaphysical battlefield is richer and more complex than a simple dichotomy between “metaphysics of science” and “traditional metaphysics”, and (...) that it should instead be understood as a three dimensional “box”, with one axis distinguishing “descriptive metaphysics” from “revisionary metaphysics”, a second axis distinguishing a priori from a posteriori metaphysics, and a third axis distinguishing “commonsense metaphysics”, “traditional metaphysics” and “metaphysics of science”. We use this three-dimensional figure to shed light on the project of current metaphysics of science, and to demonstrate that, in many instances, the target of that project is not defined with enough precision and clarity. (shrink)

In this paper, we put forward a new account of emergence called “transformational emergence”. Such an account captures a variety of emergence that can be considered as being diachronic and weakly ontological. The fact that transformational emergence actually constitutes a genuine form of emergence is motivated. Besides, the account is free of traditional problems surrounding more usual, synchronic versions of emergence, and it can find a strong empirical support in a specific physical phenomenon, the fractional quantum Hall effect, which has (...) long been touted as a paradigmatic case of emergence. (shrink)