Homotopy type theory (HoTT) enables reasoning about groups directly as the types of symmetries (automorphisms) of mathematical structures. The HoTT approach to groups—first put forward by Buchholtz, van Doorn, and Rijke—identifies a group with the type of objects of which it is the symmetries. This type is called the “delooping” of the group, taking a term from algebraic topology. This approach naturally extends the group theory to higher groups which have symmetries between symmetries, and so on. In this (...) paper, we formulate and prove a higher version of Schreier’s classification of all group extensions of a given group. Specifically, we prove that extensions of a group G by a group K are classified by actions of G on a delooping of K. Our proof is formalized in Cubical Agda, a dependently typed programming language and proof assistant which implements HoTT. (shrink)

The article introduces the concept of ‘argumentational integrity’ as the basis for developing ethical criteria by which contributions to argumentative discussions can be evaluated; the focus is on the derivation, definition, and specification of the concept. The derivation of the concept starts out from a prescriptive use of ‘argumentation’, entailing in particular the goal of a rational as well as a cooperative solution. In order to make this goal attainable, contributions to argumentative discussions must meet certain conditions. It is assumed (...) that participants are not only intuitively aware of these conditions, but in fact expect of themselves and others that they will not consciously violate the conditions. This assumption leads to the most general definition of the norm of argumentational integrity: Speakers must not knowingly violate the argumentative conditions. On the basis of an empirical study drawing upon classifications of unethical strategies in popular rhetorical texts, the general norm is then specified in the form of 11 ‘standards of fair argumentation’. (shrink)

RésuméNous introduisons la notion de structure algébrique dynamique, inspirée de l'évaluation dynamique et de la théorie des modèles. Nous montrons comment cette notion constructive permet une relecture de la théorie d'Artin-Schreier, avec la modification capitale que le résultat final est alors établi de manière constructive. Nous pensons que ce que nous avons réalisé ici sur un cas d'école peut être généralisé à des parties significatives de l'algèbre classique, et est donc une contribution à la réalisation du programme de Hilbert (...) pour l'algèbre classique.We introduce the notion of “Dynamic Algebraic Structure” inspired by Dynamic Evaluation and Model Theory. We show that this constructive notion allows a rereading of the Artin-Schreier-Robinson solution for the 17th Hilbert Problem. So, once we know how to reread the proofs, this kind of abstract theory contains an algorithm which computes the concrete result . Our method gives a constructive semantic for certain parts of abstract classical mathematics.The idea is the following: replace the classical algebraic structures “constructed” by Choice and Principle of Third Excluded Middle , by DAS and dynamic evaluations of these DAS. Then TEM is replaced by construction of branching in the trees of dynamic evaluation of the DAS. If Choice is used in the form of Godel completeness theorem, it is not really necessary to use it for obtaining concrete results: in DAS, Choice is simply replaced by … nothing!. This is because the classical proof is by contradiction: “if there were not a sum of squares then some formal theory would admit a pathological model”. The constructive reasoning is more direct: since the pathological theory proves 0 = 1 we know how to construct the sum of squares … and classical models have disappeared in the proof. They are replaced by dynamic evaluations of DAS. We think that we have given, for an academic example, a new method, realizing a kind of Hilbert Program for significant parts of classical algebra. (shrink)

We give explicit formulas witnessing IP, IP $_{\!n}$, or TP2 in fields with Artin–Schreier extensions. We use them to control p-extensions of mixed characteristic henselian valued fields, allowing us most notably to generalize to the NIP $_{\!n}$ context one way of Anscombe–Jahnke’s classification of NIP henselian valued fields. As a corollary, we obtain that NIP $_{\!n}$ henselian valued fields with NIP residue field are NIP. We also discuss tameness results for NTP2 henselian valued fields.

What kind of reality is legal reality, how is it created, and what are its a priori foundations? These are the central questions asked by the early phenomenologists who took interest in social ontology and law. While Reinach represents the well-known “realist” approach to phenomenology of law, Felix Kaufmann and Fritz Schreier belonged to the “positivist” “Vienna School of Jurisprudence,” combining Hans Kelsen’s Pure Theory of Law with Edmund Husserl’s phenomenology—and thereby challenging Reinach’s views on how legal reality (...) and the legal a priori were to be conceived. This paper addresses the controversy between these positivist and realist approaches to phenomenology of law, with the goal of introducing the lesser known theories of Kaufmann and Schreier. The special focus on their critique of Reinach’s outline should give us an overview of their positions vis-à-vis the basic and a priori elements of which legal reality consists and the role that phenomenology plays in analyzing them. There is one general tendency to be noted: While the phenomenological legal positivists see the root of legal reality in an act of interpretation according to a “normative scheme of interpretation,” Reinach locates the roots of legal reality in social interaction and argues for the existence of entities independent of any interpretation. (shrink)

The Artin‐Schreier theorem says that every formally real field has orders. Friedman, Simpson and Smith showed in [6] that the Artin‐Schreier theorem is equivalent to over. We first prove that the generalization of the Artin‐Schreier theorem to noncommutative rings is equivalent to over. In the theory of orderings on rings, following an idea of Serre, we often show the existence of orders on formally real rings by extending pre‐orders to orders, where Zorn's lemma is used. We (...) then prove that “pre‐orders on rings not necessarily commutative extend to orders” is equivalent to. (shrink)

Jan Krajíček posed the following problem: Is there is a generalization result in the theory of real closed fields of the form: If A is provable in length k for all n ϵ ω , then A is provable? It is argued that the answer to this question depends on the particular formulation of the “theory of real closed fields.” Four distinct formulations are investigated with respect to their generalization behavior. It is shown that there is a positive (...) answer to Krajíček's question for 1. the axiom system RCF of Artin-Schreier with Gentzen's LK as underlying logical calculus, 2. RCF with the variant LK B of LK allowing introduction of several quantifiers of the same type in one step, 3. LK B and the first-order schemata corresponding to Dedekind cuts and the supremum principle. A negative answer is given for 4. any system containing the schema of extensionality. (shrink)

We investigate some common points between stable structures and weakly small structures and define a structureMto befineif the Cantor-Bendixson rank of the topological space${S_\varphi }\left} \right)$is an ordinal for every finite subsetAofMand every formula$\varphi \left$wherexis of arity 1. By definition, a theory isfineif all its models are so. Stable theories and small theories are fine, and weakly minimal structures are fine. For any finite subsetAof a fine groupG, the traces on the algebraic closure$acl\left$ofAof definable subgroups ofGover$acl\left$which are boolean combinations (...) of instances of an arbitrary fixed formula can decrease only finitely many times. An infinite field with a fine theory has no additive nor multiplicative proper definable subgroups of finite index, nor Artin-Schreier extensions. (shrink)

The Physikalische Gesellschaft zu Berlin , founded in 1845 as a local association, became a centre of interdisciplinarity. Until about 1880 the Physical Society contributed to the development of interdisciplinary sciences like bio(medical)physics, astrophysics and Physics of the earth (physical meteorology). In the same way the emergence of well-founded electrotechnics and scientific device construction was promoted by physics.

Nobelprizewinner Ferdinand Braun worked as a teacher for three years (1874–1977) at the Leipzig Thomas School. In this time, essential for him, he developed his most important discovery: the effect of semiconductivity. Furthermore he demonstrated his pedagogical talent as a teacher and wrote an approval, popularized book for young people. The experiences of that time had a influence upon his future work.

This book presents fourteen new essays by international scholars about the intersections between pragmatism, education, and philosophy with children. Pragmatism from its beginnings has sought a revolution in learning, and is itself a special kind of philosophy of education. What can the applications of pragmatism to pedagogy around the world teach us today?

The Simulation Theory of Memory states that to remember an episode is to simulate it in the imagination (Michaelian, 2016a, b), making memory thus reducible to the act of imagining. This paper examines Simulation Theory’s resources to account for our ability to distinguish episodic memory from free imagination. The theory suggests that we can reliably do so because of the distinctive phenomenology episodic memory comes with (i.e., a feeling of remembering), which other episodic imaginings lack. I will (...) raise two objections to how the feeling of remembering is engineered in the theory, followed by an exhaustive exploration of the theory’s resources to ground the mechanism underlying the elicitation of such feeling. I will conclude that the Simulation Theory cannot simultaneously defend the simulational character of episodic memory and ground our ability to discriminate between memories and imaginings. (shrink)

"An important contribution to contemporary jurisprudential debate and to legal thought more generally, Just Interpretations is far ahead of currently available work."--Peter Goodrich, author of Oedipus Lex "I was struck repeatedly by the clarity of expression throughout the book. Rosenfeld's description and criticism of the recent work of leading thinkers distinguishes his work within the legal theory genre. Furthermore, his own theory is quite original and provocative."--Aviam Soifer, author of Law and the Company We Keep.

The Nielsen‐Schreier theorem asserts that subgroups of free groups are free. In the first section we show that this theorem does not follow from the Linear Ordering Principle, thus strengthening the fact that it implies the Axiom of Choice for families of finite sets. In the second section, we show that a stronger variant of the Nielsen‐Schreier theorem implies the Axiom of Choice.