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)
Wigner famously referred to the 'unreasonable effectiveness' of mathematics in its application to science. Using Wigner's own application of grouptheory 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.
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)
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 standard model of subatomic particles and the periodic table of the atoms 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 the book 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. The trail follows the lines of parentage linking groups upstream to the unitary symmetry of the eightfold way of quarks, and to the four-dimensional symmetry of the hydrogen atom. Along the way the theory of symmetry groups is gradually introduced with special emphasis on graphical representations. 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 to leave the common ground of linear algebras and explore the potential of non-linearity. (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].
This paper traces the origins of Eugene Wigner's pioneering application of grouptheory to quantum physics to his early work in chemistry and crystallography. In the early 1920s, crystallography was the only discipline in which symmetry groups were routinely used. Wigner's early training in chemistry, and his work in crystallography with Herman Mark and Karl Weissenberg at the Kaiser Wilhelm institute for fiber research in Berlin exposed him to conceptual tools which were absent from the pedagogy available to (...) physicists for many years to come. This both enabled and pushed him to apply the group theoretic approach to quantum physics. It took many years for the approach first introduced by Wigner in the 1920s – and whose reception by the physicists was initially problematical – to assume the pivotal place it now holds in physical theory and education. This is but one example that attests to the historic contribution made by the periphery in initiating new types of thought-perspectives and scientific careers. (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)
Are companies, churches, and states genuine agents? Or are they just collections of individuals that give a misleading impression of unity? This question is important, since the answer dictates how we should explain the behaviour of these entities and whether we should treat them as responsible and accountable on the model of individual agents. Group Agency offers a new approach to that question and is relevant, therefore, to a range of fields from philosophy to law, politics, and the social (...) sciences. Christian List and Philip Pettit argue that there really are group or corporate agents, over and above the individual agents who compose them, and that a proper approach to the social sciences, law, morality, and politics must take account of this fact. Unlike some earlier defences of group agency, their account is entirely unmysterious in character and, despite not being technically difficult, is grounded in cutting-edge work in social choice theory, economics, and philosophy. (shrink)
In this paper, we will prove the inevitable non-uniformity of two constructions from combinatorial grouptheory related to the word problem for finitely generated groups and the Higman—Neumann—Neumann Embedding Theorem.
This article takes a fresh look at the Ardeners' muted grouptheory, 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)
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 grouptheory, 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)
Whether upheld as heroic or reviled as terrorism, people have been willing to lay down their lives for the sake of their groups throughout history. Why? Previous theories of extreme self-sacrifice have highlighted a range of seemingly disparate factors, such as collective identity, outgroup hostility, and kin psychology. In this paper, I attempt to integrate many of these factors into a single overarching theory based on several decades of collaborative research with a range of special populations, from tribes in (...) Papua New Guinea to Libyan insurgents and from Muslim fundamentalists in Indonesia to Brazilian football hooligans. These studies suggest that extreme self-sacrifice is motivated by identity fusion, a visceral sense of oneness with the group, resulting from intense collective experiences or from perceptions of shared biology. In ancient foraging societies, fusion would have enabled warlike bands to stand united despite strong temptations to scatter and flee. The fusion mechanism has often been exploited in cultural rituals, not only by tribal societies but also in specialized cells embedded in armies, cults, and terrorist organizations. With the rise of social complexity and the spread of states and empires, fusion has also been extended to much larger groups, including doctrinal religions, ethnicities, and ideological movements. Explaining extreme self-sacrifice is not only a scientific priority but also a practical challenge as we seek a collective response to suicide, terrorism, and other extreme expressions of outgroup hostility that continue to bedevil humanity today. (shrink)
Over-time variability characterizes not only individual-level emotions, but also group-level emotions, those that occur when people identify with social groups and appraise events in terms of their implications for those groups. We discuss theory and research regarding the role of emotions in intergroup contexts, focusing on their dynamic nature. We then describe new insights into the causes and consequences of emotional dynamics that flow from conceptualizing emotions as based in group membership, and conclude with research recommendations.
Evolutionary anthropology has focused on the origins of war, or rather ethnocentricity, because it epitomizes the problem of group selection, and because war may itself have been the main agent of group selection. The neo-Darwinian synthesis in biology has explained how ethnocentricity might evolve by group selection, and the distinction between evoked culture and adopted culture, suggested by the emerging synthesis in evolutionary psychology, has explained how it might be transmitted. Ethnocentric mechanisms could have evolved by genetic (...) selection in ancestral hominids, or through the interaction of genetic and cultural selection in modern humans, or both. The existence of similar behaviors in chimpanzees and the parallel development of early human societies around the globe are arguments for such inherited mechanisms. There may have been some adaptive breakthroughs in purely cultural evolution, but this process does not seem likely to produce long-term Darwinian adaptations because of the prolificity of cultural traits. Warfare may once have been a major agent of group selection, but the rates of extinction among human groups are so slow as to render this improbable since the rise of state-level societies, whose warfare tends to become a cyclical balance-of-power situation. Perhaps the most serious implication of current evolutionary thought is that the individualistic model of culture common in the social sciences and humanities is outmoded, and should be replaced by a new model that recognizes the organismic nature of human societies. (shrink)
Analogies between classical statistical mechanics and quantum field theory played a pivotal role in the development of renormalization group methods for application in the two theories. This paper focuses on the analogies that informed the application of RG methods in QFT by Kenneth Wilson and collaborators in the early 1970's. The central task that is accomplished is the identification and analysis of the analogical mappings employed. The conclusion is that the analogies in this case study are formal analogies, (...) and not physical analogies. That is, the analogical mappings relate elements of the models that play formally analogous roles and that have substantially different physical interpretations. Unlike other cases of the use of analogies in physics, the analogical mappings do not preserve causal structure. The conclusion that the analogies in this case are purely formal carries important implications for the interpretation of QFT, and poses challenges for philosophical accounts of analogical reasoning and arguments in defence of scientific realism. Analysis of the interpretation of the cutoffs is presented as an illustrative example of how physical disanalogies block the exportation of physical interpretations from from statistical mechanics to QFT. A final implication is that the application of RG methods in QFT supports non-causal explanations, but in a different manner than in statistical mechanics. (shrink)
In both biology and the human sciences, social groups are sometimes treated as adaptive units whose organization cannot be reduced to individual interactions. This group-level view is opposed by a more individualistic one that treats social organization as a byproduct of self-interest. According to biologists, group-level adaptations can evolve only by a process of natural selection at the group level. Most biologists rejected group selection as an important evolutionary force during the 1960s and 1970s but a (...) positive literature began to grow during the 1970s and is rapidly expanding today. We review this recent literature and its implications for human evolutionary biology. We show that the rejection of group selection was based on a misplaced emphasis on genes as “replicators” which is in fact irrelevant to the question of whether groups can be like individuals in their functional organization. The fundamental question is whether social groups and other higher-level entities can be “vehicles” of selection. When this elementary fact is recognized, group selection emerges as an important force in nature and what seem to be competing theories, such as kin selection and reciprocity, reappear as special cases of group selection. The result is a unified theory of natural selection that operates on a nested hierarchy of units.The vehicle-based theory makes it clear that group selection is an important force to consider in human evolution. Humans can facultatively span the full range from self-interested individuals to “organs” of group-level “organisms.” Human behavior not only reflects the balance between levels of selection but it can also alter the balance through the construction of social structures that have the effect of reducing fitness differences within groups, concentrating natural selection at the group level. These social structures and the cognitive abilities that produce them allow group selection to be important even among large groups of unrelated individuals. (shrink)
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  and ). 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 , Ladyman , and French ). (shrink)