Hume against the Infinite Divisibility of Space

The Beckmann-Black problem asks whether any given finite Galois extension E/K of group G is the specialization at some point t 0 ∈ P 1 ( K ) of some finite regular Galois extension F / K ( T ) with the same group. In this paper, we study a generalization of this problem for infinite extensions, via the profinite twisting lemma, and apply the latter in many situations, like abelian infinite extensions and the l -universal Frattini cover of an arbitrary finite group.

We propose a statistical study of the scattering of an incident plane wave by a stack of two two-dimensional rough interfaces. The interfaces are characterized by Gaussian height distributions with zero mean values and Gaussian correlation functions. The electromagnetic fields are represented by Rayleigh expansions, and a perturbation method is used for solving the boundary value problem and determining the first-order scattering amplitudes. For slightly rough interfaces with a finite extension, we show that the modulus of the co- and cross-polarized scattering amplitudes follows a Hoyt law and that the phase is not uniformly distributed. For interfaces with an infinite extension, the modulus follows a Rayleigh law and the phase is uniformly distributed. We show that these results are true for correlated or uncorrelated interfaces and for isotropic or anisotropic interfaces.

In this chapter we shall analyze arbitrary extension fields of a field Φ. A study of finite dimensional extension fields and a partial study of algebraic extensions has been made in Chapter I. In this chapter our primary concern will be with infinite dimensional extensions and we shall begin again with the algebraic ones. We define algebraically closed fields and prove the existence of an algebraic closure of any field. We shall extend the classical Galois theory to apply to infinite dimensional normal and separable extensions. After this we shall consider arbitrary extension fields and we shall show that these can be built up in two stages: first a purely transcendental one and then on top of this an algebraic extension. The invariant of this mode of generating a field is the transcendency degree which is the cardinal number of a transcendency basis. We shall obtain conditions for the existence of a transcendency basis such that the extension is separable algebraic over the purely transcendental extension determined by the basis. We shall also give a definition of separability of an extension field that generalizes the notion of algebraic separability. The notion of a derivation plays an important role in these considerations.KeywordsPrime IdealStructure TheoryGalois GroupHigh DerivationAlgebraic ClosureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Surface and finite size effects in critical phenomena

We now apply the cohomology theory of the previous chapter to the study of the Brauer group. However, we shall have to use a slightly modified construction which takes into account the fact that the absolute Galois group of a field is determined by its finite quotients. This is the cohomology theory of profinite groups, which we develop first. As a fruit of our labours, we identify the Brauer group of a field with a second, this time commutative, cohomology group of the absolute Galois group. As applications, we obtain easy proofs of some basic facts concerning the Brauer group, and give an important characterization of the index of a central simple algebra. Last but not least, one of the main objects of study in this book makes its appearance: the Galois symbol. The cohomology theory of profinite groups was introduced in the late 1950s by John Tate, motivated by sheaf-theoretic considerations of Alexander Grothendieck. His original aim was to find the appropriate formalism for developing class field theory. Tate himself never published his work, which thus became accessible to the larger mathematical community through the famous account of Serre [4], which also contains many original contributions. It was Brauer himself who described the Brauer group as a second cohomology group, using his language of factor systems. We owe to Serre the insight that descent theory can be used to give a more conceptual proof. The Galois symbol was defined by Tate in connection with the algebraic theory of power residue symbols, a topic extensively studied in the 1960s by Bass, Milnor, Moore, Serre and others. Profinite groups and Galois groups It can be no surprise that the main application of the cohomological techniques of the previous chapter will be in the case when G is the Galois group of a finite Galois extension. However, it will be convenient to consider the case of infinite Galois extensions as well, and first and foremost that of the extension k s |k , where k s is a separable closure of k .

A contribution to the dynamical theory of radiating electrons

In this paper, I focus on a recent debate in extended cognition known as “cognitive assembly” and how cognitive assembly shares a certain kinship with the special composition question advanced in analytical metaphysics. Both the debate about cognitive assembly and the special composition question ask about the circumstances under which entities (broadly construed) compose or assemble another entity. The paper argues for two points. The first point is that insofar as the metaphysics of composition presupposes that composition is a synchronic relation of dependence, then that presupposition is inconsistent with the temporal dynamics inherent in the process of cognitive assembly. The second point is that by developing a diachronic or temporally dynamic ontology for understanding the phenomenon of cognitive assembly, this lends support for a third wave of extended cognition.

This paper proposes a novel answer to the Special Composition Question. In some respects it agrees with brutalism about composition; in others with universalism. The main novel feature of this answer is the insight I think it gives into what the debate over the Special Composition Question is about.

The Special Composition Question asks, for any xs, what are the metaphysically necessary and sufficient conditions under which there is a y such that the xs compose y. But answers to the Special Composition Question are not merely meant to provide metaphysically necessary and sufficient conditions for the fact that there is a y such that the xs compose y; answers are meant to provide conditions that explain or ground the fact that there is a y such that the xs compose y. But existential facts are grounded in their instances. So, the fact that there is a y such that the xs compose y is also grounded in its instances. But the fact that there is a y such that the xs compose y cannot be grounded both in the conditions of composition and in the instances of composition; that would constitute unacceptable overdetermination of grounding.

The special composition question (SCQ), which asks under which conditions objects compose a further object, establishes a central debate in modern metaphysics. Recent successes of inductive metaphysics, which studies the implications of the natural sciences for metaphysical problems, suggest that insights into the SCQ can be gained by investigating the physics of composite systems. In this work, I show that the minus first law of thermodynamics, which is concerned with the approach to equilibrium, leads to a new approach to the SCQ, the thermodynamic composition principle (TCP): Multiple systems in (generalized) thermal contact compose a single system. This principle, which is justified based on a systematic classification of possible mereological models for thermodynamic systems, might form the basis of an inductive argument for universalism. A formal analysis of the TCP is provided on the basis of mereotopology, which is a combination of mereology and topology. Here, “thermal contact” can be analyzed using the mereotopological predicate “self-connectedness”. Self-connectedness has to be defined in terms of mereological sums to ensure that scattered objects cannot be self-connected.

Objectives for the interactive experiences include exposure to a wide variety of computer applications; in-depth experience with at least one application; overcoming computer fear; appreciating the rigid, logical flow of computerized problem-solving; and appreciating the benefits and limitations of computer applications for a variety of purposes. Ultimately, a major purpose of transferring informatics content is related to stimulating students' imagination with respect to the computer's ability to aid their professional endeavors. Robinson (1984) describes the imagination factor: "I think the total potential of computers is only limited by our imagination. It's like giving an artist a palette that has an infinite number of colors, some of them invisible to the naked eye. . . It is a tool for the realization of ideas." Students who have been exposed to the benefits of major categories of computer applications can appreciate computer capabilities with respect to exploring scientific and nursing phenomena, and building databases to store and access information (Newbern, 1985). These students should have enough theoretical and experimental knowledge of computers to become actively involved in making creative, informed decisions about how computers will be applied to nursing in their professional setting. Hardin and Skiba (1982) noted a gap between the powerful information processing capabilities of the computer and its relatively limited use by nursing--a gap which still exists today. Students who have participated in an idea generation course on computer applications can help to bridge this gap, helping nursing to take full advantage of the computer-saturated environments of the future.

Mereological nihilism is the thesis that composition never occurs. Some philosophers have thought that science gives us compelling evidence against nihilism. In this article I respond to this concern. An initial challenge for nihilism stems from the fact that composition is such a ubiquitous feature of scientific theories. In response I motivate a restricted form of scientific anti-realism with respect to those components of scientific theories which make reference to composition. A second scientifically based worry for nihilism is that certain specific scientific phenomena (quantum entanglement, natural selection) might require ineliminable quantification over composite objects. I address these concerns, and argue that there seem to be nihilist-friendly construals of the scientific phenomena in question.

Philosophers sometimes discuss the “ontological status” of this or that kind of entity. They should be addressing one of the following questions, or the word ontological is being misused: 1. Does X exist? 2. Under what conditions can X exist? 3. Do we have good reasons to think that X exists? All three questions can be asked about elements, and have been asked. Aristotle criticized the atomist account of chemical combination, according to which elements survive in their compounds. Eighteenth-century chemists rejected the Aristotelian view, although tacitly; they simply assumed that an element lives on in its compounds. Nineteenth-century chemists gradually adopted (an adapted form of) atomism, according to which an element can exist wherever its characteristic atoms do. The periodic table also allowed them to ask, of its empty spaces, whether they correspond to real but unknown elements. Priority disputes forced them to consider when there is sufficient evidence for the discovery of new elements. In the 1920s, IUPAC proposed a very thin definition of an element, according to which an element exists wherever its characteristic nuclear charge does. But according to some scientists, it is now challenged by the fleeting existence of some superheavy elements; if a nucleus cannot survive long enough to acquire a stable electronic structure, then it cannot be said to have any chemical properties. How then can it be called a chemical element? In this paper I explore this latest ontological question, in the light of a sufficiently nuanced understanding of earlier ones. I then relate this discussion to a more general question about existence: the Special Composition Question.

The positive and negative theses of Ladyman and Ross's Every Thing Must Go are clarified and defended, and exemplified by how composition is treated in metaphysics and in science. Section 2 provides an apology for the book in two parts, the first presenting the book's negative content concerning analytic metaphysics, and the second developing its positive metaphysics. Section 3 exemplifies the negative and positive theses of ETMG by criticizing the debate about van Inwagen's special composition question, and outlining how composition in condensed matter and particle physics is in accord with the real patterns account of existence in which scale relativity and objective modality are central.

Let $F$ be a finite extension field of the rational numbers, $Q$, and let $K$ be an infinite abelian extension of $F$. Let $S$ be a finite set of prime divisors of $Q$ including the Archimedean one. An $S$-unit of $K$ is a field element which is a local unit at all prime divisors of $F$ which do not restrict on $Q$ to a member of $S$. It is shown that the group of $S$-units of $K$ is the direct product of the group of roots of unity of $K$ with a free abelian group.