There is currently much interest in bringing together the tradition of categorial grammar, and especially the Lambek calculus, with the recent paradigm of linear logic to which it has strong ties. One active research area is designing non-commutative versions of linear logic (Abrusci, 1995; Retoré, 1993) which can be sensitive to word order while retaining the hypothetical reasoning capabilities of standard (commutative) linear logic (Dalrymple et al., 1995). Some connections between the Lambek calculus and computations in groups have long been (...) known (van Benthem, 1986) but no serious attempt has been made to base a theory of linguistic processing solely on group structure. This paper presents such a model, and demonstrates the connection between linguistic processing and the classical algebraic notions of non-commutative free group, conjugacy, and group presentations. A grammar in this model, or G-grammar is a collection of lexical expressions which are products of logical forms, phonological forms, and inverses of those. Phrasal descriptions are obtained by forming products of lexical expressions and by cancelling contiguous elements which are inverses of each other. A G-grammar provides a symmetrical specification of the relation between a logical form and a phonological string that is neutral between parsing and generation modes. We show how the G-grammar can be oriented for each of the modes by reformulating the lexical expressions as rewriting rules adapted to parsing or generation, which then have strong decidability properties (inherent reversibility). We give examples showing the value of conjugacy for handling long-distance movement and quantifier scoping both in parsing and generation. The paper argues that by moving from the free monoid over a vocabulary V (standard in formal language theory) to the free group over V, deep affinities between linguistic phenomena and classical algebra come to the surface, and that the consequences of tapping the mathematical connections thus established can be considerable. (shrink)

Symmetry is at the heart of our understanding of matter. This book tells the fascinating story of the constituents of matter from a common symmetry perspective. The standard model of elementary particles and the periodic table of chemical elements have the common goal to bring order in the bewildering chaos of the constituents of matter. Their success relies on the presence of fundamental symmetries in their core. -/- The purpose of Shattered Symmetry is to share the admiration for the power (...) and the beauty of these symmetries. The reader is taken on a journey from the basic geometric symmetry group of a circle to the sublime dynamic symmetries that govern the motions of the particles. Along the way the theory of symmetry groups is gradually introduced with special emphasis on its use as a classification tool and its graphical representations. This is applied to the unitary symmetry of the eightfold way of quarks, and to the four-dimensional symmetry of the hydrogen atom. The final challenge is to open up the structure of Mendeleev's table which goes beyond the symmetry of the hydrogen atom. Breaking this symmetry to accommodate the multi-electron atoms requires us to leave the common ground of linear algebras and explore the potential of non-linearity. (shrink)

We review some of the progress made in the past 27 years in understanding the group theoretic and path integral aspects of the hydrogen atom. The group theoretic development was triggered by A. O. Barut who suggested to me the search for a dynamical group larger than SO(4). In this way he became indirectly responsible also for important recent path integral developments.

In this paper we investigate a number of analytical solutions to the polynomial class of nonlinear Klein-Gordon equations in multidimensional spacetime. This is done in the context of classical φ4 and φ6 field theory, the former with and without the inclusion of an external force field conjugate to φ. Both massive (m≠0) and massless (m=0) cases are considered, as well as tachyonic solutions allowed (v>c). We first present a complete set of translationally invariant solutions for the φ4 model and (...) demonstrate the role of external force fields in altering the form of these solutions. Next, spherically symmetric solutions are discussed in both φ4 and φ6 cases since they provide the most realistic models of elementary particles. (shrink)

The commentary is in general agreement with Roger Shepard's view of evolutionary internalization of certain procedural memories, but advocates the use of Lie groups to express the invariances of motion and color perception involved. For categorization, the dialectical pair is suggested. [Barlow; Hecht; Kubovy & Epstein; Schwartz; Shepard; Todorovic].

To this day, a hundred and fifty years after Mendeleev's discovery, the overal structure of the periodic system remains unaccounted for in quantum-mechanical terms. Given this dire situation, a handful of scientists in the 1970s embarked on a quest for the symmetries that lie hidden in the periodic table. Their goal was to explain the table's structure in group-theoretical terms. We argue that this symmetry program required an important paradigm shift in the understanding of the nature of chemical elements. (...) The idea, in essence, consisted of treating the chemical elements, not as particles, but as states of a superparticle. We show that the inspiration for this came from elementary particle physics, and in particular from Heisenberg's suggestion to treat the proton and neutron as different states of the nucleon. We provide a careful study of Heisenberg's last paper on the nature of elementary particles, and explain why the Democritean picture of matter no longer applied in modern physics and a Platonic symmetry-based picture was called for instead. We show how Heisenberg's Platonic philosophy came to dominate the field of elementary particle physics, and how it found its culmination point in Gell-Mann's classification of the hadrons in the eightfold way. We argue that it was the success of Heisenberg's approach in elementary particle physics that sparked the group-theoretical approach to the periodic table. We explain how it was applied to the set of chemical elements via a critical examination of the work of the Russian mathematician Abram Ilyich Fet the Turkish-American physicist Asim Orhan Barut, before giving some final reflections. (shrink)

This article takes a fresh look at the Ardeners' muted group theory, originally applied in social anthropology and later taken up by the women's movement. The theory has wider applicability in aiding understanding of the communication processes between females and males but there is a need for a combination of disparate types of research extending the focus beyond mutedness as a structural product to the processes by which women are rendered mute, involving a broader analysis of the (...) political, economic and organizational context. The communications research arena is envisaged as one in which further work could be carried out. (shrink)

This book is the first to systematically develop, explain, and apply diagrammatic projection operators to construct all semi-simple Lie algebras, both classical and exceptional.

Wigner famously referred to the `unreasonable effectiveness' of mathematics in its application to science. Using Wigner's own application of group theory to nuclear physics, I hope to indicate that this effectiveness can be seen to be not so unreasonable if attention is paid to the various idealising moves undertaken. The overall framework for analysing this relationship between mathematics and physics is that of da Costa's partial structures programme.

Introduction. This survey is intended to introduce to logicians some notions, methods and theorems in set theory which arose—largely through the work of Saharon Shelah—out of attempts to solve problems in abelian group theory, principally the Whitehead problem and the closely related problem of the existence of almost free abelian groups. While Shelah's first independence result regarding the Whitehead problem used established set-theoretical methods, his later work required new ideas; it is on these that we focus. We (...) emphasize the nature of the new ideas and the historical context in which they arose, and we do not attempt to give precise technical definitions in all cases, nor to include a comprehensive survey of the algebraic results.In fact, very little algebraic background is needed beyond the definitions of group and group homomorphism. Unless otherwise specified, we will use the word “group” to refer to an abelian group, that is, the group operation is commutative. The group operation will be denoted by +, the identity element by 0, and the inverse of a by −a. We shall use na as an abbreviation for a + a + … + a [n times] if n is positive, and na = if n is negative. (shrink)

A. F. Bentley’s The Process of Government (1908) is widely accepted as an important source of contemporary interest group study. This paper argues to the contrary that Bentley’s arguments in this area are obscure and have contributed little to the programme of modern interest group research. His importance is as a contributor to the debate on the nature of social science and social science method and not as the starting-point for interest group analysis. The judgement about his (...) role as a social scientist should rest on consideration of his body of work and not simply the one book. In terms of his much cited book, Bentley, it is argued, is misread. The central purpose of this article is to explore the consequences of that misinterpretation. The misreading of The Process of Government, and the unmerited assumption that it is directly connected to modern interest group theory, has led to a misunderstanding of that contemporary theory. In particular his use of the term ‘group’ is much wider in scope than is now usually followed. This means that his claims are not so uni-dimensional as they appear when extracted from their context. Bentley used the term in a sociological sense that included informal social associations as ‘groups’: these are not the sort of formal, collective organizations of the interest group type as identified in political science. It is argued that the major sources of ideas current in the interest group field are Truman (1951) and the case-study authors of the 1930s such as Odegard, Childs, Herring and Schattschneider. Bentley’s contribution to political science is not as progenitor of interest group studies, but his emphasis on process anticipates the policy studies movement. (shrink)

This four volume set is dedicated to the work of Emile Durkheim, one of the most important and prolific sociologists in the field, who is commonly cited as a founding father of modern social science. With volumes published between 1975 and 1991, this collection brings together a range of modern critical responses to Durkheim's work across a broad range of topics, including: epistemology, modernism and post-modernism, theories of social order, and the rise and development of modern society. The authors in (...) the collection also draw important comparisons between Durkheim and other seminal sociologists, including Max Weber and Claude Bernard. _For institutional purchases for e-book sets please contact [email protected]. (shrink)

Reissuing works originally published between 1949 and ‘79, this set presents a rich selection of renowned scholarship across the subject, touching also on ethics, religion, and psychology and other behavioural science. Classic previously out-of-print works are brought back into print here in this set of important discourse and theory.

This 20 volume Routledge Revivals collection brings together a selection of groundbreaking Philosophy titles, from the rich and diverse Routledge backlist. With titles published between 1933 and 1991, this is a truly wide-ranging selection, encompassing works by distinguished authors such as: Simone Weil, Hilary Putnam, Franz Brentano, Anthony Kenny, Karl Jaspers and Israel Scheffler. Dealing with everything from the notion of freewill, to concepts of time and space, to theories of morality, this set offers a collection of the best of (...) Routledge publishing in the field of Philosophy from across the Twentieth Century. Please note that all titles have been previously available for sale individually through the Routledge Revivals programme. (shrink)

We develop the theory of germs of generic functions in simple theories. Starting with an algebraic quadrangle (or other similar hypotheses), we obtain an "almost" generic group chunk, where the product is denned up to a bounded number of possible values. This is the first step towards the proof of the group configuration theorem for simple theories, which is completed in [3].

We achieve several results. First, we develop a variant of the theory of absolute Galois groups in the context of many sorted structures. Second, we provide a method for coding absolute Galois groups of structures, so they can be interpreted in some monster model with an additional predicate. Third, we prove the “Weak Independence Theorem” for pseudo-algebraically closed (PAC) substructures of an ambient structure with no finite cover property (nfcp) and the property [Formula: see text]. Fourth, we describe Kim-dividing (...) in these PAC substructures and show several results related to the SOPnhierarchy. Fifth, we characterize the algebraic closure in PAC structures. (shrink)

We study model theory of fields with actions of a fixed finite group scheme. We prove the existence and simplicity of a model companion of the theory of such actions, which generalizes our previous results about truncated iterative Hasse–Schmidt derivations [13] and about Galois actions [14]. As an application of our methods, we obtain a new model complete theory of actions of a finite group on fields of finite imperfection degree.

We study hyperdefinable groups, the most general kind of groups interpretable in a simple theory. After developing their basic theory, we prove the appropriate versions of Hrushovski's group quotient theorem and the Weil–Hrushovski group chunk theorem. We also study locally modular hyperdefinable groups and prove that they are bounded-by-Abelian-by-bounded. Finally, we analyze hyperdefinable groups in supersimple theories.

In this paper, we will prove the inevitable non-uniformity of two constructions from combinatorial group theory related to the word problem for finitely generated groups and the Higman—Neumann—Neumann Embedding Theorem.

In this paper, I shall discuss the heuristic role of symmetry in the mathematical formulation of quantum mechanics. I shall first set out the scene in terms of Bas van Fraassen’s elegant presentation of how symmetry principles can be used as problem-solving devices (see van Fraassen [1989] and [1991]). I will then examine in what ways Hermann Weyl and John von Neumann have used symmetry principles in their work as a crucial problem-solving tool. Finally, I shall explore one consequence of (...) this situation to recent debates about structural realism (SR) and empiricism in physics (Worrall [1989], Ladyman [1998], and French [1999]). (shrink)

We don't yet have adequate theories of what the human mind is representing when it represents a social group. Worse still, many people think we do. This mistaken belief is a consequence of the state of play: Until now, researchers have relied on their own intuitions to link up the conceptsocial groupon the one hand and the results of particular studies or models on the other. While necessary, this reliance on intuition has been purchased at a considerable cost. When (...) looked at soberly, existing theories of social groups are either (i) literal, but not remotely adequate (such as models built atop economic games), or (ii) simply metaphorical (typically a subsumption or containment metaphor). Intuition is filling in the gaps of an explicit theory. This paper presents a computational theory of what, literally, a group representation is in the context of conflict: It is the assignment of agents to specific roles within a small number of triadic interaction types. This “mental definition” of a group paves the way for acomputational theory of social groups– in that it provides a theory of what exactly the information-processing problem of representing and reasoning about a group is. For psychologists, this paper offers a different way to conceptualize and study groups, and suggests that a non-tautological definition of a social group is possible. For cognitive scientists, this paper provides a computational benchmark against which natural and artificial intelligences can be held. (shrink)

Klein’s Erlangen program contains the postulate to study thegroup of automorphisms instead of a structure itself. This postulate, takenliterally, sometimes means a substantial loss of information. For example, thegroup of automorphisms of the field of rational numbers is trivial. Howeverin the case of Euclidean plane geometry the situation is different. We shallprove that the plane Euclidean geometry is mutually interpretable with theelementary theory of the group of authomorphisms of its standard model.Thus both theories differ practically in the language (...) only. (shrink)

Groups are ubiquitous in our lives. But while some of them are highly structured and appear to support a shared intentionality and even a shared agency, others are much less cohesive and do not seem to demand much of their individual members. Queues, for example, seem to be, at a given time, nothing over and above some individuals as they exemplify a certain spatial arrangement. Indeed, the main aim of this paper is to develop the more general thought that at (...) a given time, a group is nothing over and above some individual members as they exemplify a certain complex condition. The general conception of groups that emerges is able to accommodate a variety of constraints on a reasonable answer to the question of what are groups. (shrink)

Let G be a finite group. We explore the model-theoretic properties of the class of differential fields of characteristic zero in m commuting derivations equipped with a G-action by differential fie...

I propose a `stage theory’ of groups: a group is a fusion of group-stages, where a group-stage is a plurality of individuals at a world and a time. The stage theory consists of existence conditions, identity conditions, and parthood conditions for groups.

Philosophy typically ignores biographical, historical, and cultural aspects of theorists' lives in an attempt to take a supposedly abstract and objective view of their work. This book makes some new conclusions about Arendt’s theory by emphasizing how her experience of the world as displayed in her archival materials impacted her thought. Some aspects of Arendt’s life have been examined in detail before, including the fact she was stateless as well as her affair with Heidegger. Instead, this work explores different (...) topics including the biographical and narrative moments of Arendt's own work, the role of archiving in her thought, pivotal events that have not been archived, her understanding of her own identities, and how it affected the role of identity politics in her work. Typically, group action is underemphasized in Arendt scholarship in comparison to individual action and often identity politics questions are considered to lie within the realm of the private. Although Arendt’s theory is problematic when discussing issues concerning identity politics, she did think identity politics could be public and political and that effective political actions may occur within groups. What makes this project unique are the innovative conclusions made by moving the archival and biographical evidence to the center in order to understand her theory more accurately and within its historical and cultural context. (shrink)

La sapienza di partire da sè è una via che si sottrae alle molte contrapposizioni che sono iscritte nel simbolico dominante: quella tra soggettivo e oggettivo, tra individuo e comunità e tra locale e generale: apre un'altra strada.

This paper presents a modified random network model to illustrate how groups can form in the absence of evolutionary forces, assuming groups are collections of entities at any level of organization. This model is inspired by the Zero Force Evolutionary Law, which states that there is always a tendency for diversity and complexity to increase in any evolutionary system containing variation and heredity. That is, in the absence of evolutionary forces, the expectation is a continual increase in diversity and complexity (...) at any level of biological hierarchy. I show that, when modeled, this expectation of increasing variation results not only in the formation of groups, but also in a higher probability of group formation than is found in a model that is purely random. (shrink)

This is a survey, intended both for group theorists and model theorists, concerning the structure of pseudofinite groups, that is, infinite models of the first-order theory of finite groups. The focus is on concepts from stability theory and generalisations in the context of pseudofinite groups, and on the information this might provide for finite group theory.

the theory of modal groups is an abstract theory that may be used to show formal similarities between various formulations of epistemic logics (eg Hintikka, Chisholm, and von Wright) and formal systems in other domains.