We prove that the theory IΔ0, extended by a weak version of the Δ0-Pigeonhole Principle, proves that every integer is the sum of four squares (Lagrange's theorem). Since the required weak version is derivable from the theory IΔ0 + ∀x (xlog(x) exists), our results give a positive answer to a question of Macintyre (1986). In the rest of the paper we consider the number-theoretical consequences of a new combinatorial principle, the ‘Δ0-Equipartition Principle’ (Δ0EQ). In particular we give a new (...) proof, which can be formalized in IΔ0 + Δ0EQ, of the fact that every prime of the form 4n + 1 is the sum of two squares. (shrink)

The symplectic structures (brackets, Hamilton's equations, and Lagrange's equations) for the Dirac electron and its classical model have exactly the same form. We give explicitly the Poisson brackets in the dynamical variables (x μ,p μ,v μ,S μv). The only difference is in the normalization of the Dirac velocities γμγμ=4 which has significant consequences.

Recent work on the history of General Relativity by Renn, Sauer, Janssen et al. shows that Einstein found his field equations partly by a physical strategy including the Newtonian limit, the electromagnetic analogy, and energy conservation. Such themes are similar to those later used by particle physicists. How do Einstein's physical strategy and the particle physics derivations compare? What energy-momentum complex did he use and why? Did Einstein tie conservation to symmetries, and if so, to which? How did his work (...) relate to emerging knowledge of the canonical energy-momentum tensor and its translation-induced conservation? After initially using energy-momentum tensors hand-crafted from the gravitational field equations, Einstein used an identity from his assumed linear coordinate covariance x'=Mx to relate it to the canonical tensor. Usually he avoided using matter Euler-Lagrange equations and so was not well positioned to use or reinvent the Herglotz-Mie-Born understanding that the canonical tensor was conserved due to translation symmetries, a result with roots in Lagrange, Hamilton and Jacobi. Whereas Mie and Born were concerned about the canonical tensor's asymmetry, Einstein did not need to worry because his Entwurf Lagrangian is modeled not so much on Maxwell's theory as on a scalar theory. Einstein's theory thus has a symmetric canonical energy-momentum tensor. But as a result, it also has 3 negative-energy field degrees of freedom. Thus the Entwurf theory fails a 1920s-30s a priori particle physics stability test with antecedents in Lagrange's and Dirichlet's stability work; one might anticipate possible gravitational instability. This critique of the Entwurf theory can be compared with Einstein's 1915 critique of his Entwurf theory for not admitting rotating coordinates and not getting Mercury's perihelion right. One can live with absolute rotation but cannot live with instability. Particle physics also can be useful in the historiography of gravity and space-time, both in assessing the growth of objective knowledge and in suggesting novel lines of inquiry to see whether and how Einstein faced the substantially mathematical issues later encountered in particle physics. This topic can be a useful case study in the history of science on recently reconsidered questions of presentism, whiggism and the like. Future work will show how the history of General Relativity, especially Noether's work, sheds light on particle physics. (shrink)

Schwinger's action principle is formulated for the quantum system which corresponds to the classical system described by the LagrangianL c( $\dot x$ , x)=(M/2)gij(x) $\dot x$ i $\dot x$ j−v(x). It is sufficient for the purpose of deriving the laws of quantum mechanics to consider onlyc-number variations of coordinates and time. The Euler-Lagrange equation, the canonical commutation relations, and the canonical equations of motion are derived from this principle in a consistent manner. Further, it is shown that an arbitrary (...) point transformation leaves the forms of the fundamental equations invariant. The judicious choice of the quantal Lagrangian is essential in our formulation. A quantum mechanical analog of Noether's theorem, which relates the invariance of the quantal action with a conservation law, is established. The ambiguities in the quantal Lagrangian are also discussed and it is pointed out that the requirement of invariance is not sufficient to determine uniquely the quantal Lagrangian and the Hamiltonian. (shrink)

'The Matrix Project' comprises two contexts, namely photographic work of architect Thierry Lagrange and the self-reflective trajectory that he developed during his doctorate at KULeuven. Lagrange's practice-based research developed to an autonomous artistic project, in which he studied the signification and nature of observation within a creative process. Through his multifaceted practice as an architect, photographer and researcher, Lagrange developed 'The Matrix Method'; a method that stimulates creativity within different contexts and can be implemented for example in (...) art education. For the Matrix Project, 25 years' worth of photographs was entered into a context of research and presents a new perspective on different ways of seeing in our present visual culture. (shrink)

We are confronted by the clash of contradictory ideologies and a crisis of universal knowledge. Two major causes of this crisis are the erosion of common sense and a relativistic view of doctrinal development. Fr. Réginald Garrigou-Lagrange foresaw today's crisis and wrote keenly in defense of the classical Thomistic synthesis. His critiques of modern philosophy and theology, we are now able to see, were prophetic. This first-time English translation of his Le sens commun: La philosophie de l'être et les (...) formules dogmatiques addresss the relationship between common sense and both philosophy and faith; the proper defense for philosophical realism; the subordination and coordination of philosophical first principles; our natural capacity for knowing God's existence; and, at length, the problem of dogmatic development. (shrink)

Cognitive Science, Volume 46, Issue 4, April 2022.

Objectives The objectives of the study are to understand the current practice of informed consent in medical research in public hospitals in Shanghai, and to share our views with other countries, especially developing countries. Methods In the study, 145 consent forms (CFs) of the selected research projects in eight public hospitals with ethics committees in Shanghai were audited, and the principle investigators (PIs) of these research projects and 40 student subjects who had participated in clinical drug tests were surveyed by (...) questionnaires. Results The CFs of medical researches in public hospitals with ethics committees in Shanghai were generally acceptable. However, there were some defects in the CFs. Although most of the surveyed PIs had correct recognition of informed consent, some processes of informed consent were not in accordance with generally accepted requirements. A large number of the PIs considered the greatest difficulty with informal consent was lack of correct recognition of subjects or legally authorised representatives on medical research. Discussion Informed consent in medical research should consider the research ethics, the background of potential subjects, the local resources and culture of medical research. In addition, special protection is needed for student subjects in informed consent as well as efforts for building and restoring the public's trust in biomedical research. Conclusion The informed consent in Shanghai's public hospitals with ethics committees was generally acceptable and the achievement of adequate informed consent is influenced by many factors. (shrink)

Critiqués pour leur manque d’objectivité et de rationalité, les témoignages d’observations d’ovnis et d’« ufonautes » invitent-ils, dans le contexte de la crise écologique, à reposer la question de la pluralité des mondes extraterrestres ou à ouvrir plutôt celle de la pluralité des mondes terrestres? Invitent-ils à s’interroger sur les preuves de la présence d’E. T. dans notre environnement ou sur la nécessité de cesser de voir le monde dans lequel nous évoluons comme un stock de « ressources », afin (...) d’entrer en contact avec cette pluralité d’êtres et y trouver notre place dans un monde débarrassé du partage entre nature et culture? (shrink)

This text is an exploration of the metaphysical principle, "Every agent acts for an end." It is split into two parts, the first being primarily pedagogical and general, the second topical. In the first part, Fr. Garrigou-Lagrange sets forth the basics of the Aristotelian metaphysics of teleology, defending its place as a central point of metaphysics. After defending its per se nota character, he summarizes a number of main corollaries to the principle, primarily within the perspective established by traditional (...) Thomistic accounts of metaphysics, doing so in a way that is pedagogically sensitive yet speculatively profound. In the second half, he gathers together a number of articles which he had written, each having some connection with themes concerning teleology. Thematically, the texts consider the finality and teleology of the human intellect and will, along with the way that the principle of finality sheds light on certain problems associated with the distinction between faith and reason. Finally, the text ends with an important essay on the principle of the mutual interdependence of causes, causae ad invicem sunt causae, sed in diverso genere. (shrink)