In this paper, we consider certain cardinals in ZF (set theory without AC, the axiom of choice). In ZFC (set theory with AC), given any cardinals C and D, either C ≤ D or D ≤ C. However, in ZF this is no longer so. For a given infinite set A consider $\operatorname{seq}^{1 - 1}(A)$ , the set of all sequences of A without repetition. We compare $|\operatorname{seq}^{1 - 1}(A)|$ , the cardinality of this set, to |P(A)|, the cardinality of (...) the power set of A. What is provable about these two cardinals in ZF? The main result of this paper is that $ZF \vdash \forall A(|\operatorname{seq}^{1 - 1}(A)| \neq|\mathscr{P}(\mathscr{A})|)$ , and we show that this is the best possible result. Furthermore, it is provable in ZF that if B is an infinite set, then $|\operatorname{fin}(B)| <|\mathscr{P}(B)|$ even though the existence for some infinite set B* of a function f from $\operatorname{fin}(B^\ast)$ onto P(B*) is consistent with ZF. (shrink)

If we assume the axiom of choice, then every two cardinal numbers are comparable, In the absence of the axiom of choice, this is no longer so. For a few cardinalities related to an arbitrary infinite set, we will give all the possible relationships between them, where possible means that the relationship is consistent with the axioms of set theory. Further we investigate the relationships between some other cardinal numbers in specific permutation models and give some results provable without using (...) the axiom of choice. (shrink)

For \(n\in \omega \), the weak choice principle \(\textrm{RC}_n\) is defined as follows: _For every infinite set_ _X_ _there is an infinite subset_ \(Y\subseteq X\) _with a choice function on_ \([Y]^n:=\{z\subseteq Y:|z|=n\}\). The choice principle \(\textrm{C}_n^-\) states the following: _For every infinite family of_ _n_-_element sets, there is an infinite subfamily_ \({\mathcal {G}}\subseteq {\mathcal {F}}\) _with a choice function._ The choice principles \(\textrm{LOC}_n^-\) and \(\textrm{WOC}_n^-\) are the same as \(\textrm{C}_n^-\), but we assume that the family \({\mathcal {F}}\) is linearly orderable (...) (for \(\textrm{LOC}_n^-\) ) or well-orderable (for \(\textrm{WOC}_n^-\) ). In the first part of this paper, for \(m,n\in \omega \) we will give a full characterization of when the implication \(\textrm{RC}_m\Rightarrow \textrm{WOC}_n^-\) holds in \({\textsf {ZF}}\). We will prove the independence results by using suitable Fraenkel-Mostowski permutation models. In the second part, we will show some generalizations. In particular, we will show that \(\textrm{RC}_5\Rightarrow \textrm{LOC}_5^-\) and that \(\textrm{RC}_6\Rightarrow \textrm{C}_3^-\), answering two open questions from Halbeisen and Tachtsis (Arch Math Logik 59(5):583–606, 2020). Furthermore, we will show that \(\textrm{RC}_6\Rightarrow \textrm{C}_9^-\) and that \(\textrm{RC}_7\Rightarrow \textrm{LOC}_7^-\). (shrink)

For a set M, let \\) denote the set of all finite sequences which can be formed with elements of M, and let \ denote the set of all 2-element subsets of M. Furthermore, for a set A, let Open image in new window denote the cardinality of A. It will be shown that the following statement is consistent with Zermelo–Fraenkel Set Theory \: There exists a set M such that Open image in new window and no function Open image (...) in new window is finite-to-one. (shrink)

For an integer \, Ramsey Choice\ is the weak choice principle “every infinite setxhas an infinite subset y such that\ has a choice function”, and \ is the weak choice principle “every infinite family of n-element sets has an infinite subfamily with a choice function”. In 1995, Montenegro showed that for \, \. However, the question of whether or not \ for \ is still open. In general, for distinct \, not even the status of “\” or “\” is known. (...) In this paper, we provide partial answers to the above open problems and among other results, we establish the following:1.For every integer \, if \ is true for all integers i with \, then \ is true for all integers i with \.2.If \ are any integers such that for some prime p we have \ and \, then in \: \ and \.3.For \, \\\ implies \, and \ implies neither \ nor \ in \.4.For every integer \, \ implies “every infinite linearly orderable family of k-element sets has a partial Kinna–Wagner selection function” and the latter implication is not reversible in \ ). In particular, \ strictly implies “every infinite linearly orderable family of 3-element sets has a partial choice function”.5.The Chain-AntiChain Principle implies neither \ nor \ in \, for every integer \. (shrink)

In this article we give a forcing characterization for the Ramsey property of Σ 1 2 -sets of reals. This research was motivated by the well-known forcing characterizations for Lebesgue measurability and the Baire property of Σ 1 2 -sets of reals. Further we will show the relationship between higher degrees of forcing absoluteness and the Ramsey property of projective sets of reals.

For a ⊆ b ⊆ ω with b\ a infinite, the set D = {x ∈ [ω]ω : a ⊆ x ⊆ b} is called a doughnut. A set S ⊆ [ω]ω has the doughnut property [MATHEMATICAL SCRIPT CAPITAL D] if it contains or is disjoint from a doughnut. It is known that not every set S ⊆ [ω]ω has the doughnut property, but S has the doughnut property if it has the Baire property ℬ or the Ramsey property ℛ. (...) In this paper it is shown that a finite support iteration of length ω1 of Cohen forcing, starting from L, yields a model for CH + equation image + equation image + equation image. (shrink)

We prove versions of the Dual Ramsey Theorem and the Dual Ellentuck Theorem for families of partitions which are defined in terms of games.

In this article we compare the well-known Ramsey property with a dual form of it, the so called dual-Ramsey property (which was suggested first by Carlson and Simpson). Even if the two properties are different, it can be shown that all classical results known for the Ramsey property also hold for the dual-Ramsey property. We will also show that the dual-Ramsey property is closed under a generalized Suslin operation (the similar result for the Ramsey property was proved by Matet). Further (...) we compare two notions of forcing, the Mathias forcing and a dual form of it, and will give some symmetries between them. Finally we give some relationships between the dual-Mathias forcing and the dual-Ramsey property. (shrink)

Preface to Volumes 1 and 2 Lorenz Krüger xv Introduction to Volume 1 Lorraine J. Daston 1 I Revolution 1 What Are Scientific Revolutions? Thomas S. Kuhn 7 2 Scientific Revolutions, Revolutions in Science, and a Probabilistic Revolution 1800-1930 I. Bernard Cohen 23 3 Was There a Probabilistic Revolution 1800-1930? Ian Hacking 45 II Concepts 4 The Slow Rise of Probabilism: Philosophical Arguments in the Nineteenth Century Lorenz Krüger 59 5 The Decline of the Laplacian Theory of Probability: (...) A Study of Stumpf, von Kries, and Meinong Andreas Kamlah 91 6 Fechner’s Indeterminism: From Freedom to Laws of Chance Michael Heidelberger 117 7 The Saint Petersburg Paradox 1713-1937 Gerard Jorland 157 8 Laplace and Thereafter: The Status of Probability Calculus in the Nineteenth Century Ivo Schneider 191 9 Emile Borel as a Probabilist Eberhard Knobloch 215 III Uncertainty 10 The Domestication of Risk: Mathematical Probability and Insurance 1650-1830 237 11 The Objectification of Observation: Measurement and Statistical Methods in the Nineteenth Century Zeno G. Swijtink 261 12 The Measurement of Uncertainty in Nineteenth-Century Social Science Stephen M. Stigler 287 IV Society 13 Rational Individuals versus Laws of Society: From Probability to Statistics Lorraine J. Daston 295 14 Decrire, Compter, Calculer: The Debate over Statistics during the Napoleonic Period Marie-Noelle Bourguet 305 15 Probability in Vital and Social Statistics: Quetelet, Farr, and the Bertillons Bernard-Pierre Lécuyer 317 16 Paupers and Numbers: The Statistical Argument for Social Reform in Britain during the Period of Industrialization Karl H. Metz 337 17 Lawless Society: Social Science and the Reinterpretation of Statistics in Germany, 1850-1880 Theodore M. Porter 351 18 Prussian Numbers 1860-1882 Ian Hacking 377 19 How Do Sums Count? On the Cultural Origins of Statistical Causality M. Norton Wise 395. (shrink)

With the outbreak of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), global researchers were confronted with major challenges. The German National Pandemic Cohort Network (NAPKON) was launched in fall 2020 to effectively leverage resources and bundle research activities in the fight against the coronavirus disease 2019 (COVID-19) pandemic. We analyzed the setup phase of NAPKON as an example for multicenter studies in Germany, highlighting challenges and optimization potential in connecting 59 university and nonuniversity study sites. We examined the ethics application (...) process of 121 ethics submissions considering durations, annotations, and outcomes. Study site activation and recruitment processes were investigated and related to the incidence of SARS-CoV-2 infections. For all initial ethics applications, the median time to a positive ethics vote was less than two weeks and 30 of these study sites (65%) joined NAPKON within less than three weeks each. Electronic instead of postal ethics submission (9.5 days (Q1: 5.75, Q3: 17) vs. 14 days (Q1: 11, Q3: 26), p value = 0.01) and adoption of the primary ethics vote significantly accelerated the ethics application process. Each study center enrolled a median of 37 patients during the 14-month observation period, with large differences depending on the health sector. We found a positive correlation between recruitment performance and COVID-19 incidence as well as hospitalization incidence. Our analysis highlighted the challenges and opportunities of the federated system in Germany. Digital ethics application tools, adoption of a primary ethics vote and standardized formal requirements lead to harmonized and thus faster study initiation processes during a pandemic. (shrink)

This piece presents the work of academics and architects in a collaborative venture. It provides an architectural design and a series of statements towards the hypothetical creation of an unconventional city centre in the Chinese city of Shenzhen. The idea is to create a linear university that would run the 20-kilometer length of the Shenzhen Strip: the 20K university. The contributors outline, in the diversity of their idioms, a complex spatial condition fundamental to life, and demonstrate new relationships between knowledge (...) and the city. The design of the proposed ‘open university space’ responds to two simultaneous and interrelated challenges: that posed to architecture, and that posed to science. The university would embody the meeting of these at the intersection of the urban infrastructure and the knowledge infrastructure. The purpose is thus also to develop the notion of knowledge, embodied in institutions, as urban infrastructure. (shrink)

The main purpose of the paper is to investigate the relevance and significance of the concept of common good in contemporary society. First, I make a brief historical remark about the philosophical concept of common good. I will argue that the concept is rooted in the ancient Greek philosophical understanding of society, namely as polis, whereby human being is thought to have an end that is not merely individual but also collective. I then discuss how societies have significantly changed over (...) the years and how the current global order resembles the situation during the time of Alexander the Great, whose vision it was to establish a cosmopolis, literally a global city. In the end, I consider whether the notion of common good in itself has lost its relevance in the face of the manifold social changes. I bring my discussion to a close with a note on the universality and naturality of the common good of humankind. (shrink)

Pleasure has always been an important issue in morality. And although ethical systems tend to focus the discussion on human action, this agreeable sentiment has remained a recurrent question in moral philosophy. In this paper, I go back to Aristotle’s treatment of pleasure in his writings, particularly in the Nicomachean Ethics. I will argue that the distinction he draws between bodily pleasures and those of the mind represents an important point not only in understanding eudaimonia but also in situating the (...) very nature of ethical debates today. I will try to show how Aristotle’s distinction among pleasure matches up with his differentiation between energeia and kinésis as well as with the distinction between praxis and poiésis, and how these differentiations have enabled Aristotle toevade the pitfall of hedonism. (shrink)

POHLMAN S, CIBULKA NJ, PALMER JL, LORENZ RA and SMITHBATTLE L. Nursing Inquiry 2013; 20: 71–81 The placebo puzzle: examining the discordant space between biomedical science and illness/healingThe placebo response presents an enigma to biomedical science: how can ‘inert’ or ‘sham’ procedures reduce symptoms and produce physiological changes that are comparable to prescribed treatments? In this study, we examine this puzzle by explicating the discordant space between the prevailing biomedical paradigm, which focuses on a technical understanding of diagnosis and (...) treatment, and a broader understanding of illness and healing as relational and embodied. Although biomedical achievements are impressive, the knowledge resulting from this paradigm is limited by its ontological and epistemological assumptions. When the body and world are objectified, illness meanings, therapeutic relationships, and healing practices are dismissed or distorted. In spite of a robust critique of the tenets of biomedicine for guiding practice, the biomedical paradigm retains a tenacious hold on evidence-based medicine and nursing, downplaying our clinical understanding of the sentient body, patients’ life-worlds, and illness and healing. In reality, skilled nurses rely on multiple forms of knowledge in providing high-quality care to particular patients. Clinically wise nurses integrate their experience and knowledge of patients’ priorities, fears, and illness trajectories along with biomedical findings to make astute judgments and promote health and healing. (shrink)

Health care decision making for patients without decisional capacity is ethically and legally challenging. Advance directives (living wills) have proved to be of limited usefulness in clinical practice. Therefore, academic attention should focus more on substitute decision making by the next of kin. In this article, we comparatively analyse the legal approaches to substitute medical decision making in England and Germany. Based on the current ethico-legal discourse in both countries, three aspects of substitute decision making will be highlighted: (1) Should (...) there be a legally predefined order of relatives who serve as health care proxies? (2) What should be the respective roles and decisional powers of patient-appointed versus court-appointed substitute decision-makers? (3) Which criteria should be determined by law to guide substitute decision-makers? (shrink)

Die "Gottesfrage" ist anerkanntermaßen eine äußerst komplexe, vieldeutige und obskure Frage. In ausführlicher Auseinandersetzung mit Heidegger, in dessen Schriften eine sehr einflussreiche einseitige und entstellende Interpretation und Kritik der großen metaphysischen Tradition immer wieder präsentiert wird, zeigt Lorenz B. Puntel, dass die Gottesfrage klar, rational und argumentativ, kurz: mit theoretischer Rigorosität, behandelt werden kann. Dies aber setzt einen umfassenden philosophischen Theorierahmen voraus, der die Entfaltung einer systematischen Konzeption des Seins als solchen und im Ganzen ermöglicht und in nuce enthält. (...) Anschließend wird eine solche Konzeption im Grundriss zur Darstellung gebracht. In einem ausführlichen Kapitel werden die Positionen von É. Lévinas und J.-L. Marion, den zwei wohl radikalsten postmodernen Kritikern und Gegnern jeder Form eines positiven Verhältnisses zwischen dem "Sein(sdenken)" und (dem Thema) "Gott", eingehend analysiert und deren radikale Unhaltbarkeit demonstriert. (shrink)

Die "Gottesfrage" ist anerkanntermaßen eine äußerst komplexe, vieldeutige und obskure Frage. In ausführlicher Auseinandersetzung mit Heidegger, in dessen Schriften eine sehr einflussreiche einseitige und entstellende Interpretation und Kritik der großen metaphysischen Tradition immer wieder präsentiert wird, zeigt Lorenz B. Puntel, dass die Gottesfrage klar, rational und argumentativ, kurz: mit theoretischer Rigorosität, behandelt werden kann. Dies aber setzt einen umfassenden philosophischen Theorierahmen voraus, der die Entfaltung einer systematischen Konzeption des Seins als solchen und im Ganzen ermöglicht und in nuce enthält. (...) Anschließend wird eine solche Konzeption im Grundriss zur Darstellung gebracht. In einem ausführlichen Kapitel werden die Positionen von É. Lévinas und J.-L. Marion, den zwei wohl radikalsten postmodernen Kritikern und Gegnern jeder Form eines positiven Verhältnisses zwischen dem "Sein" und "Gott", eingehend analysiert und deren radikale Unhaltbarkeit demonstriert. (shrink)

I PSYCHOLOGY 5 The Probabilistic Revolution in Psychology--an Overview Gerd Gigerenzer 7 1 Probabilistic Thinking and the Fight against Subjectivity Gerd Gigerenzer 11 2 Statistical Method and the Historical Development of Research Practice in American Psychology Kurt Danziger 35 3 Survival of the Fittest Probabilist: Brunswik, Thurstone, and the Two Disciplines of Psychology Gerd Gigerenzer 49 4 A Perspective for Viewing the Integration of Probability Theory in Psychology David J. Murray 73 II SOCIOLOGY 101 5 The Two Empirical Roots of (...) Social Theory and the Probability Revolution Anthony Oberschall 103 III ECONOMICS 133 The Probabilistic Revolution in Economics--an Overview Mary S. Morgan 135 6 Why Was There No Probabilistic Revolution in Economic Thought? Claude Mènard 139 7 The Rise of Macroeconomic Calculations in Economic Statistics Robert A. Horváth 147 8 Statistics without Probability and Haavelmo’s Revolution in Econometrics Mary S. Morgan 171 IV PHYSIOLOGY 199 9 Experimental Physiology and Statistical Inference: The Therapeutic Trial in Nineteenth-Century Germany William Coleman 201 V EVOLUTIONARY BIOLOGY 227 The Probabilistic Revolution in Evolutionary Biology--an Overview John H. Beatty 229 10 Natural Selection as a Causal, Empirical, and Probabilistic Theory M. J. S. Hodge 233 11 Dobzhansky and Drift: Facts, Values, and Chance in Evolutionary Biology John H. Beatty 271 12 Random Genetic Drift, R. A. Fisher, and the Oxford School of Ecological Genetics John R. G. Turner 313 13 On the Prior Probability of the Existence of Life Bernd-Olaf Küppers 355 VI PHYSICS 371 The Probabilistic Revolution in Physics--an Overview Lorenz Krüger 373 14 Probabilistic Physics the Classical Way Jan von Plato 379 15 Max Born and the Reality of Quantum Probabilities Nancy Cartwright 409 16 Philosophical Problems of Quantum Theory: The Response of American Physicists Nancy Cartwright. (shrink)

This paper studies Aumann’s agreeing to disagree theorem from the perspective of dynamic epistemic logic. This was first done by Dégremont and Roy (J Phil Log 41:735–764, 2012) in the qualitative framework of plausibility models. The current paper uses a probabilistic framework, and thus stays closer to Aumann’s original formulation. The paper first introduces enriched probabilistic Kripke frames and models, and various ways of updating them. This framework is then used to prove several agreement theorems, which are natural formalizations of (...) Aumann’s original result. Furthermore, a sound and complete axiomatization of a dynamic agreement logic is provided, in which one of these agreement theorems can be derived syntactically. These technical results are used to show the importance of explicitly representing the dynamics behind the agreement theorem, and lead to a clarification of some conceptual issues surrounding the agreement theorem, in particular concerning the role of common knowledge. The formalization of the agreement theorem thus constitutes a concrete example of the so-called dynamic turn in logic. (shrink)

The allocation of any scarce health care resource, especially a lifesaving resource, can create profound ethical and legal challenges. Liver transplant allocation currently is based upon urgency, a sickest-first approach, and does not utilize capacity to benefit. While urgency can be described reasonably well with the MELD system, benefit encompasses multiple dimensions of patients’ well-being. Currently, the balance between both principles is ill-defined. This survey with 502 participants examines how urgency and benefit are weighted by different stakeholders. Liver transplant patients (...) favored the sickest-first allocation, although all other groups tended to favor benefit. Criteria of a successful transplantation were a minimum survival of at least 1 year and recovery of functional status to being ambulatory and capable of all self-care. An individual delisting decision was accepted when the 1-year survival probability would fall below 50%. Benefit was found to be a critical variable that may also trigger the willingness to donate organs. The strong interest of stakeholder for successful liver transplants is inadequately translated into current allocation rules. (shrink)

Background The allocation of any scarce health care resource, especially a lifesaving resource, can create profound ethical and legal challenges. Liver transplant allocation currently is based upon urgency, a sickest-first approach, and does not utilize capacity to benefit. While urgency can be described reasonably well with the MELD system, benefit encompasses multiple dimensions of patients’ well-being. Currently, the balance between both principles is ill-defined. Methods This survey with 502 participants examines how urgency and benefit are weighted by different stakeholders. Results (...) Liver transplant patients favored the sickest-first allocation, although all other groups tended to favor benefit. Criteria of a successful transplantation were a minimum survival of at least 1 year and recovery of functional status to being ambulatory and capable of all self-care. An individual delisting decision was accepted when the 1-year survival probability would fall below 50%. Benefit was found to be a critical variable that may also trigger the willingness to donate organs. Conclusions The strong interest of stakeholder for successful liver transplants is inadequately translated into current allocation rules. (shrink)

Erwin Panofsky explicitly states that the first half of the opening chapter of Studies in Iconology—his landmark American publication of 1939—contains ‘the revised content of a methodological article published by the writer in 1932’, which is now translated for the first time in this issue of Critical Inquiry.1 That article, published in the philosophical journal Logos, is among his most important works. First, it marks the apogee of his series of philosophically reflective essays on how to do art history,2 that (...) reach back, via a couple of major pieces on Alois Riegl, to the 1915 essay on Heinrich Wölfflin.3 Under the influence of his colleague at Hamburg Ernst Cassirer, the principal interpreter of Kant in the 1920s, Panofsky from 1915 on exhibits in his work ever more Kantian thinking and language.4 But Logos was not an art-historical review or one dedicated to aesthetics but a principal mainstream journal of the philosophy of culture. So ‘On the Problem of Describing and Interpreting Works of the Visual Arts’ has a good claim to be the culmination of Panofsky's philosophical thinking in his German period under the Weimar Republic. · 1. Erwin Panofsky, Studies in Iconology: Humanistic Themes in the Art of the Renaissance , p. xv; hereafter abbreviated SI. See Panofsky, ‘Zum Problem der Beschreibung und Inhaltsdeutung von Werken der bildenden Kunst’, Logos 21 : 103–19; trans. Jaś Elsner and Katharina Lorenz under the title ‘On the Problem of Describing and Interpreting Works of the Visual Arts’, Critical Inquiry 38 : 467–82; hereafter abbreviated ‘P’.· 2. See the discussion in Carlo Ginzburg, ‘From Aby Warburg to E. H. Gombrich: A Problem of Method’, Myths, Emblems, Clues, trans. John and Anne C. Tedeschi , pp. 17–59, esp. pp. 36–41.· 3. See Panofsky, ‘Das Problem des Stils in der bildenden Kunst’, Deutschsprachige Aufsätze, ed. Karen Michels and Martin Warnke, 2 vols. , 2:1009–18; ‘Der Begriff des Kunstwollens’,Deutschsprachige Aufsätze, 2:1019–34, trans. Kenneth J. Northcott and Joel Snyder under the title ‘The Concept of Artistic Volition’, Critical Inquiry 8 : 17–33; and ‘Über das Verhältnis der Kunstgeschichte zur Kunsttheorie: Ein Beitrag zu der Erörterung über die Möglichkeit kunstwissenschaftlicher Grundbegriffe’, Deutschsprachige Aufsätze, 2: 1035–63, trans. Lorenz and Elsner under the title ‘On the Relationship of Art History and Art Theory: Towards the Possibility of a Fundamental System of Concepts for a Science of Art’, Critical Inquiry 35 : 43–71.· 4. On neo-Kantianism in pre-Nazi Germany, see Michael Friedman, A Parting of the Ways: Carnap, Cassirer, and Heidegger , pp. 25–37; Éric Dufour and T. Z. R. Créteil, ‘Le Statue du singulier: Kant et le néokantisme de l’École de Marbourg', Kantstudien 93 : 324–50; Edward Skidelsky, Ernst Cassirer: The Last Philosopher of Culture , pp. 22–51; and Peter E. Gordon, Continental Divide: Heidegger, Cassirer, Davos , pp. 52–86. Specifically on the Cassirerian Kantianism of Panofsky, see Michael Podro, The Critical Historians of Art , pp. 181–82; Michael Ann Holly, Panofsky and the Foundations of Art History , pp. 91–92, 147–52; Silvia Ferretti, Cassirer, Panofsky, and Warburg: Symbol, Art, and History, trans. Richard Pierce , pp. 174–77, 182–84; David Summers, ‘Meaning in the Visual Arts as a Humanistic Discipline’, in Meaning in the Visual Arts: Views from the Outside, ed. Irving Lavin , pp. 9–24; Mark A. Cheetham, Kant, Art, and Art History: Moments of Discipline , pp. 68–77; Paul Crowther, The Transhistorical Image: Philosophizing Art and Its History , pp. 70–73; Allister Neher, ‘“The Concept of Kunstwollen”, Neo-Kantianism, and Erwin Panofsky's Early Art Theoretical Essays', Word and Image 20 : 41–51; Georges Didi-Huberman,Confronting Images: Questioning the Ends of a Certain History of Art, trans. John Goodman , pp. 4–6, 90–138; and Lorenz and Elsner, ‘Translators’ Introduction', Critical Inquiry35 : 33–42, esp. pp. 38, 40–42. (shrink)

September 2008 Abstract: We consider the precursors to the discovery of sensitive dependence on initial conditions by Edward Lorenz (1963) in his model of climatic fluid dynamics. This will focus on work in various disciplines that imply either such sensitivity, irregular endogenous dynamic patterns, or fractal nature of an attractor, as is also found in the attractor underlying the model Lorenz studied. Going from ancient hints in Anaxagoras through nineteenth century mathematics and physics, the main areas of such (...) development will be argued to have been in celestial mechanics, oscillators, and economics. (shrink)

The European Legacy, Volume 17, Issue 4, Page 548-549, July 2012.

Using covariant derivatives and the operator definitions of quantum mechanics, gauge invariant Proca and Lehnert equations are derived and the Lorenz condition is eliminated in U(1) invariant electrodynamics. It is shown that the structure of the gauge invariant Lehnert equation is the same in an O(3) invariant theory of electrodynamics.

In September 2008, 10 years after the untimely death of Pere Alberch (1954–1998), the 20th Altenberg Workshop in Theoretical Biology gathered a group of Pere’s students, col- laborators, and colleagues (Figure 1) to celebrate his contribu- tions to the origins of EvoDevo. Hosted by the Konrad Lorenz Institute for Evolution and Cognition Research (KLI) outside Vienna, the group met for two days of discussion. The meeting was organized in tandem with a congress held in May 2008 at the Cavanilles (...) Institute for Biodiversity and Evolutionary Biology (ICBiBE) in Valencia, Spain. The talks at the KLI were equal parts: nostalgic remembrance, excitement over new ways of thinking about old problems, and an unrepressed vitriol against the resurgence of reductionist thinking in EvoDevo. Here we highlight some of the key aspects of Pere’s life and work that informed and infused the talks. (shrink)