We show that any simple group of Morley rank 5 is a bad group all of whose proper definable connected subgroups are nilpotent of rank at most 2. The main result is then used to catalog the nonsoluble connected groups of Morley rank 5.

Let T be a complete, superstable theory with fewer than ${2^{\aleph_{0}}}$ countable models. Assuming that generic types of infinite, simple groups definable in T eq are sufficiently non-isolated we prove that ω ω is the strict upper bound for the Lascar rank of T.

In his testamentary letter to Auguste Chevalier, Évariste Galois states that, in modern terminology, the smallest simple group has order 60. No proof of this statement survives in his papers, and it has been suggested that a proof would have been impossible using the methods available at the time. We argue that this assertion is unduly pessimistic. Moreover, one fragmentary document, dismissed as a triviality and misunderstood, looks suspiciously like cryptic notes related to this result. We give an elementary (...) proof of Galois’s statement, explain why it is likely that he would have been aware of the methods involved, and discuss the potential relevance of the fragment. (shrink)

In this paper we apply social epistemology to mathematical proofs and their role in mathematical knowledge. The most famous modern collaborative mathematical proof effort is the Classification of Finite Simple Groups. The history and sociology of this proof have been well-documented by Alma Steingart (2012), who highlights a number of surprising and unusual features of this collaborative endeavour that set it apart from smaller-scale pieces of mathematics. These features raise a number of interesting philosophical issues, but have received (...) very little attention. In this paper, we will consider the philosophical tensions that Steingart uncovers, and use them to argue that the best account of the epistemic status of the Classification Theorem will be essentially and ineliminably social. This forms part of the broader argument that in order to understand mathematical proofs, we must appreciate their social aspects. (shrink)

Simple Heuristics That Make Us Smart invites readers to embark on a new journey into a land of rationality that differs from the familiar territory of cognitive science and economics. Traditional views of rationality tend to see decision makers as possessing superhuman powers of reason, limitless knowledge, and all of eternity in which to ponder choices. To understand decisions in the real world, we need a different, more psychologically plausible notion of rationality, and this book provides it. It is (...) about fast and frugal heuristics--simple rules for making decisions when time is pressing and deep thought an unaffordable luxury. These heuristics can enable both living organisms and artificial systems to make smart choices, classifications, and predictions by employing bounded rationality. But when and how can such fast and frugal heuristics work? Can judgments based simply on one good reason be as accurate as those based on many reasons? Could less knowledge even lead to systematically better predictions than more knowledge? Simple Heuristics explores these questions, developing computational models of heuristics and testing them through experiments and analyses. It shows how fast and frugal heuristics can produce adaptive decisions in situations as varied as choosing a mate, dividing resources among offspring, predicting high school drop out rates, and playing the stock market. As an interdisciplinary work that is both useful and engaging, this book will appeal to a wide audience. It is ideal for researchers in cognitive psychology, evolutionary psychology, and cognitive science, as well as in economics and artificial intelligence. It will also inspire anyone interested in simply making good decisions. (shrink)

The notion of ecological rationality implies that the accuracy of a decision strategy depends on features of the information environment in which it is tested. We demonstrate that the performance of a group may be strongly affected by the decision strategies used by its individual members and specify how this effect is moderated by environmental features. Specifically, in a set of simulation studies, we systematically compared four decision strategies used by the individual group members: two linear, compensatory decision strategies and (...) two simple, noncompensatory heuristics. Individual decisions were aggregated by using a majority rule. To assess the ecological rationality of the strategies, we varied (a) the distribution of cue validities, (b) the quantity, and (c) the quality of shared information. Group performance strongly depended on the distribution of cue validities. When validities were linearly distributed, groups using a compensatory strategy achieved the highest accuracy. Conversely, when cue validities followed a J-shaped distribution, groups using a simple lexicographic heuristic performed best. While these effects were robust across different quantities of shared information, the quality of shared information exerted stronger effects on group performance. Consequences for prescriptive theories on group decision making are discussed. (shrink)

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.

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 talk about groups as doing something, we talk as if groups have agency. Is our talk legitimate? Are there group agents? Is there something like group agency? In this paper, I discuss two ontological frameworks concerning existence questions: the Quinean framework and the Thomasson-Carnap framework. I apply them to the problem of group agency. I review the Quinean-oriented literature debating the existence of group agents and its methodological background. I argue, via Thomasson’s easy approach to ontology, that (...) deflationism can simplify the debate surrounding group agents. Thus, I argue for a Thomasson-Carnap framework and show that it is better suited to answer the particular question whether there are group agents. More specifically, I argue for a non-reductive simple realist view on group agents, i.e. I argue for the truth of “There are group agents,” via analytic entailments, by truths about the actions and deeds of groups. (shrink)

We lay down the groundwork for the treatment of almost hyperdefinable groups: notions from [5] are put into a natural hierarchy, and new notions, essential to the study to such groups, fit elegantly into this hierarchy. We show that "classical" properties of definable and hyperdefinable groups in simple theories can be generalised to this context. In particular, we prove the existence of stabilisers of Lascar strong types and of the connected and locally connected components of subgroups, (...) and that in a simple one-based theory an almost hyperdefinable group is bounded-by-abelian-by-bounded. (shrink)

We continue the study of simple theories begun in [3] and [5]. We first find the right analogue of definability of types. We then develop the theory of generic types and stabilizers for groups definable in simple theories. The general ideology is that the role of formulas (or definability) in stable theories is replaced by partial types (or ∞-definability) in simple theories.

In recent work, the authors have established the group configuration theorem for simple theories, as well as some of its main applications from geometric stability theory, such as the binding group theorem, or in the $\omega$-categorical case, the characterization of the forking geometry of a finitely based non-trivial locally modular regular type as projective geometry over a finite field and the equivalence of pseudolinearity and local modularity. The proof necessitated an extension of the model-theoretic framework to include almost hyperimaginaries, (...) and the study of polygroups. (shrink)

We consider the investigation of the embedding of semigroups in groups, a problem which spans the early-twentieth-century development of abstract algebra. Although this is a simple problem to state, it has proved rather harder to solve, and its apparent simplicity caused some of its would-be solvers to go awry. We begin with the analogous problem for rings, as dealt with by Ernst Steinitz, B. L. van der Waerden and Øystein Ore. After disposing of A. K. Sushkevich’s erroneous contribution (...) in this area, we present A. I. Maltsev’s example of a cancellative semigroup which may not be embedded in a group, which showed for the first time that such an embedding is not possible in general. We then look at the various conditions that were derived for such an embedding to take place: the sufficient conditions of Paul Dubreil and others, and the necessary and sufficient conditions obtained by A. I. Maltsev, Vlastimil Pták and Joachim Lambek. We conclude with some comments on the place of this problem within the theory of semigroups, and also within abstract algebra more generally. (shrink)

We generalize tools and results from first order stable theories to groups inside a simple stable strongly homogeneous model.

We show that if p is a real type which is almost internal in a formula φ in a simple theory, then there is a type p' interalgebraic with a finite tuple of realizations of p, which is generated over φ. Moreover, the group of elementary permutations of p' over all realizations of φ is type-definable.

We show that any simple group of Morley rank 4 must be a bad group with no proper definable subgroups of rank larger than 1. We also give an application to groups acting on sets of Morley rank 2.

In the basic model of Condorcet’s jury theorem and in the literature that follows, an odd-numbered group of voters is assumed so that the simple majority rule can be used. We show that this assumption is not necessary either in Condorcet’s basic model or in the general framework of dichotomous choice. We first apply simple majority rule to an even-numbered homogeneous fixed-size committee. We then provide a justification for using simple majority rule for an even-numbered heterogeneous fixed-size (...) committee when the competence structure of the committee members is not common knowledge. (shrink)

We reconstruct the group action in the group configuration theorem. We apply it to show that in an ω-categorical theory a finitely based pseudolinear regular type is locally modular, and the geometry associated to a finitely based locally modular regular type is projective geometry over a finite field.

In a simple theory with elimination of finitary hyperimaginaries if tp(a) is real and analysable over a definable set Q, then there exists a finite sequence ( $a_{i}|i \leq n^{*}$ ) $\subseteq dcl^{eq}$ (a) with $a_{n}*$ = a such that for every $i \leq n*$ , if $p_{i} = tp(a_{i}/{a_{i}|j < i}$ ) then $Aut(p_{i}/Q)$ is type-definable with its action on $p_{i}^{c}$ . A unidimensional simple theory eliminates the quantifier $\exists^{\infty}$ and either interprets (in $C^{eq}$ ) an infinite (...) type-definable group or has the property that ACL(Q) = C for every infinite definable set Q. (shrink)

The binding group theorem for stable theories is partially extended to the simple context. Some results concerning internality are proved. We also introduce a ‘small’ normal subgroup G0+ of the automorphism group and show that if p is Q-internal then it has a finite exponent and G/G0+ is interpretable.

We deal with two forms of the "uniqueness cases" in the classification of large simple K*-groups of finite Morley rank of odd type, where large means the 2-rank m2 is at least three. This substantially extends results known for even larger groups having Prüfer 2-rank at least three, so as to cover the two groups PSp 4 and G 2. With an eye towards more distant developments, we carry out this analysis for L*-groups, a context (...) which is substantially broader than the K* setting. (shrink)

Let T be simple, work in Ceq over a boundedly closed set. Let p ∈ S(θ) be internal in a quasi-stably-embedded type-definable set Q (e.g., Q is definable or stably-embedded) and suppose (p, Q) is ACL-embedded in Q (see definitions below). Then Aut(p/Q) with its action on pC is type-definable in Ceq over θ. In particular, if p ∈ S(θ) is internal in a stably-embedded type-definable set Q, and pC υ Q is stably-embedded, then Aut(p/Q) is type-definable with its (...) action on pC. (shrink)

In this paper, we approach the idea of group cognition from the perspective of the “extended mind” thesis, as a special case of the more general claim that systems larger than the individual human, but containing that human, are capable of cognition (Clark, 2008; Clark & Chalmers, 1998). Instead of deliberating about “the mark of the cognitive” (Adams & Aizawa, 2008), our discussion of group cognition is tied to particular cognitive capacities. We review recent studies of group problem-solving and group (...) memory which reveal that specific cognitive capacities that are commonly ascribed to individuals are also aptly ascribed at the level of groups. These case studies show how dense interactions among people within a group lead to both similarity-inducing and differentiating dynamics that affect the group's ability to solve problems. This supports our claim that groups have organization-dependent cognitive capacities that go beyond the simple aggregation of the cognitive capacities of individuals. Group cognition is thus an emergent phenomenon in the sense of Wimsatt (1986). We further argue that anybody who rejects our strategy for showing that cognitive properties can be instantiated at multiple levels in the organizational hierarchy on a priori grounds is a “demergentist,” and thus incurs the burden of proof for explaining why cognitive properties are “stuck” at a certain level of organizational structure. Finally, we show that our analysis of group cognition escapes the “coupling-constitution” charge that has been leveled against the extended mind thesis (Adams & Aizawa, 2008). (shrink)

This paper completes the proof of the group configuration theorem for simple theories started in [1]. We introduce the notion of an almost hyperdefinable structure, and show that it has a reasonable model theory. We then construct an almost hyperdefinable group from a polygroup chunk.

We show how the notion of full Frobenius group of finite Morley rank generalizes that of bad group, and how it seems to be more appropriate when we consider the possible existence (still unknown) of nonalgebraic simple groups of finite Morley rank of a certain type, notably with no involution. We also show how these groups appear as a major obstacle in the analysis of FT-groups, if one tries to extend the Feit-Thompson theorem to groups (...) of finite Morley rank. (shrink)

If a group is modelled as a single Bayesian agent, what should its beliefs be? I propose an axiomatic model that connects group beliefs to beliefs of group members, who are themselves modelled as Bayesian agents, possibly with different priors and different information. Group beliefs are proven to take a simple multiplicative form if people’s information is independent, and a more complex form if information overlaps arbitrarily. This shows that group beliefs can incorporate all information spread over the individuals (...) without the individuals having to communicate their (possibly complex and hard-to-describe) private information; communicating prior and posterior beliefs sufices. JEL classification: D70, D71.. (shrink)

Dance provides natural conditions for studying relationships between coordination patterns and human experience. von Zimmermann and colleagues investigate whether relatively simple or more complex forms of movement coordination are related to pro‐social experiences during group dance. They find that pro‐social experience depends on the degree to which movement patterns are distributed and diversified, but not the degree to which movement patterns are simply unitary.

SKOS (Simple Knowledge Organization System) is a recommendation from the World Wide Web Consortium (W3C) for representing controlled vocabularies, taxonomies, thesauri, classifications, and similar systems for organizing and indexing information as linked data elements in the Semantic Web, using the Resource Description Framework (RDF). The SKOS data model is centered on “concepts”, which can have preferred and alternate labels in any language as well as other metadata, and which are identified by addresses on the World Wide Web (URIs). Concepts (...) are grouped into hierarchies through “broader” and “narrower” relations, with “top concepts” at the broadest conceptual level. Concepts are also organized into “concept schemes”, also identified by URIs. Other relations, mappings, and groupings are also supported. This article discusses the history of the development of SKOS and provides notes on adoption, uses, and limitations. (shrink)

Group polarization occurs when an initial attitude or belief of individuals becomes more radical after group discussion. Polarization often leads subgroups towards opposite directions. Since the 1960s this effect has been observed and repeatedly confirmed in lab experiments by social psychologists. Persuasive Arguments Theory emerged as the most convincing explanation for this phenomenon. This paper is a first attempt to frame the PAT explanation more formally by means of Bipolar Argumentation Frameworks. In particular, I show that polarization may emerge in (...) a BAF by simple and rational belief updates by participants. (shrink)

We know quite a lot about the general structure of ω-stable solvable centerless groups of finite Morley rank. Abelian groups of finite Morley rank are also well-understood. By comparison, nonabelian nilpotent groups are a mystery except for the following general results:• An ω1-categorical torsion-free nonabelian nilpotent group is an algebraic group over an algebraically closed field of characteristic 0 [Z3].• A nilpotent group of finite Morley rank is the central product of a definable subgroup of finite exponent (...) and of a definable divisible subgroup [N3].• A divisible nilpotent group of finite Morley rank is the direct product of its torsion part (which is central) and of a torsion-free subgroup [N3].However, we do not understand nilpotent groups of bounded exponent. It seems that the classification of nilpotent (but nonabelian)p-groups of finite Morley rank is impossible. Even the nilpotent groups of Morley rank 2 contain insurmountable difficulties [C], [T]. At first glance, this may seem to be an obstacle to proving the Cherlin-Zil'ber conjecture (“simple groups of finite Morley rank are algebraic groups”). Our purpose in this article is to show that if such a group is a definable subgroup of a nonnilpotent group, then it is possible to obtain a classification within the boundaries of our present knowledge. In this respect, our article may be considered as a relief to those who are trying to classify simple groups of finite Morley rank.Before explicitly stating our result, we need the following definition. (shrink)

A group is small if it has only countably many complete n-types over the empty set for each natural number n. More generally, a group G is weakly small if it has only countably many complete 1-types over every finite subset of G. We show here that in a weakly small group, subgroups which are definable with parameters lying in a finitely generated algebraic closure satisfy the descending chain conditions for their traces in any finitely generated algebraic closure. An infinite (...) weakly small group has an infinite abelian subgroup, which may not be definable. A small nilpotent group is the central product of a definable divisible group with a definable one of bounded exponent. In a group with simple theory, any set of pairwise commuting elements is contained in a definable finite-by-abelian subgroup. First corollary: a weakly small group with simple theory has an infinite definable finite-by-abelian subgroup. Secondly, in a group with simple theory, a solvable group A of derived length n is contained in an A-definable almost solvable group of class at most 2n — 1. (shrink)

Public debate on acceptable farm animal husbandry suffers from a confusion of tongues. To clarify positions of various stakeholder groups in their joint search for acceptable solutions, the concept of animal welfare was split up into three notions: no suffering, respect for intrinsic value, and non-appalling appearance of animals. This strategy was based on the hypothesis that multi-stakeholder solutions should be based on shared values rather than on compromises. The usefulness of such an artificial value distinction strategy was tested (...) in a small series of experiments. The results demonstrate that the chosen concept to distinguish between values is effective in a stakeholder context. Farmers’ views on doing good to animals appeared to be largely based on their value to prevent suffering and predominantly focused on the provision of regular care. Their priority for this value is clearly shared with other stakeholders, providing a basis for joint solutions. The concept of intrinsic value does not play a discernable role in farmers’ considerations. Based on the varying views on welfare, it can be inferred that there is a gradual rather than a principal difference between government legislation and farmers’ values, whereas public perception and acceptance of farm practices remains complicated. Distinction between value groups and focusing on a selected notion (such as no suffering) proved to be effective in bringing representatives of stakeholder groups together, but is unlikely to bridge the emotional gap between commercial farm practices and public ideals. (shrink)

An analysis of group justification enables us to understand what it means to say that a research group is justified in making a claim on the basis of evidence. I defend Frederick Schmitt's (1994) joint account of group justification by arguing against a simple summative account of group justification. Also, I respond to two objections to the joint account, one claiming that social epistemologists should always prefer the epistemic value of making true judgments to the epistemic value of maintaining (...) consistency, and another one claiming that the notion of joint commitment implicit in the joint account is epistemically unacceptable. (shrink)

We show that R.J. Thompson’s groupsFandTare bi-interpretable with the ring of the integers. From a result by A. Khélif, these groups are quasi-finitely axiomatizable and prime. So, the groupTprovides an example of a simple group which is quasi-finitely axiomatizable and prime. This answers questions posed by T. Altınel and A. Muranov in [2], and by A. Nies in [12].

Background Previous reports have described that simple cognitive training using reading aloud and solving simple arithmetic calculations, so-called “learning therapy”, can improve executive functions and processing speed in the older adults. Nevertheless, it is not well-known whether learning therapy improve a wide range of cognitive functions or not. We investigated the beneficial effects of learning therapy on various cognitive functions in healthy older adults. Methods We used a single-blinded intervention with two groups (learning therapy group: LT and (...) waiting list control group: WL). Sixty-four elderly were randomly assigned to LT or WL. In LT, participants performed reading Japanese aloud and solving simple calculations training tasks for 6 months. WL did not participate in the intervention. We measured several cognitive functions before and after 6 months intervention periods. Results Compared to WL, results revealed that LT improved inhibition performance in executive functions (Stroop: LT (Mean = 3.88) vs. WL (Mean = 1.22), adjusted p =.013 and reverse Stroop LT (Mean = 3.22) vs. WL (Mean = 1.59), adjusted p =.015), verbal episodic memory (logical memory: LT (Mean = 4.59) vs. WL (Mean = 2.47), adjusted p =.015), focus attention(D-CAT: LT (Mean = 2.09) vs. WL (Mean = -0.59), adjusted p =.010) and processing speed compared to the waiting list control group (digit symbol coding: LT (Mean = 5.00) vs. WL (Mean = 1.13), adjusted p =.015 and symbol search: LT (Mean = 3.47) vs. WL (Mean = 1.81), adjusted p =.014). Discussion This RCT can showed the benefit of learning therapy on inhibition of executive functions, verbal episodic memory, focus attention, and processing speed in healthy elderly people. Our results were discussed under overlapping hypothesis. Trial registration “This trial was registered in The University Hospital Medical Information Network Clinical Trials Registry (UMIN000006998).”. (shrink)

Tarski [19] proved the important theorem that the class of representable relation algebras is equationally axiomatizable. One of the key steps in his proof is showing that the class of (isomorphs of) simple set relation algebras—that is, algebras of binary relations with a unit of the formU×Ufor some non-empty setU—is universal, i.e., is axiomatizable by a set of universal sentences. In the same paper Tarski observed that the class of (isomorphs of) relation algebras constructed from groups (so-calledgroup relation (...) algebras) is also universal.We shall abstract the essential ingredients of Tarski's method (in Corollary 2.4), and then combine them with some observations about atom structures, to establish (in Theorem 2.6) a rather general method for showing that certain classes of simple relation algebras—and, more generally, certain classes of simple algebras in a discriminator variety V—are universal, and consequently that the collections of (isomorphs of) subdirect products of algebras in such classes form subvarieties of V. As applications of the method we show that two well-known classes of simple relation algebras, those constructed from projective geometries (sometimes calledLyndon algebras) and those constructed from modular lattices with a zero (sometimes calledMaddux algebras), are universal. In the process we prove that these two classes consist precisely of all (isomorphs of) complex algebras over the respective geometries and modular lattices, provided that we choose the primitive notions of the latter structures in an appropriate fashion. We also derive Tarski's theorems and a related theorem of the author as easy corollaries of Theorem 2.6. (shrink)

Gigerenzer, Todd, and the ABC Research Group argue that optimisation under constraints leads to an infinite regress due to decisions about how much information to consider when deciding. In certain cases, however, their fast and frugal heuristics lead instead to an endless series of decisions about how best to decide.

The Cherlin–Zil'ber Conjecture states that all simple groups of finite Morley rank are algebraic. We prove that any minimal counterexample to this conjecture has a unique conjugacy class of Carter subgroups, which are analogous to Cartan subgroups in algebraic groups.

This commentary focuses on three issues raised by Gigerenzer, Todd, and the ABC Research Group (1999). First, I stress the need for further experimental evidence to determine which heuristics people use in cognitive judgment tasks. Second, I question the scope of cognitive models based on simple heuristics, arguing that many aspects of cognition are too sophisticated to be modeled in this way. Third, I note the complementary role that rational explanation can play to Gigenerenzer et al.'s “ecological” analysis of (...) why heuristics succeed. (shrink)

Given a simple non-trivial finite-dimensional Lie algebra L, fields $K_i$ and Chevalley groups $L(K_i)$ , we first prove that $\Pi_{\mathcal{U}} L(K_i)$ is isomorphic to $L(\Pi_{\mathcal{U}}K_i)$ . Then we consider the case of Chevalley groups of twisted type ${}^n\!L$ . We obtain a result analogous to the previous one. Given perfect fields $K_i$ having the property that any element is either a square or the opposite of a square and Chevalley groups ${}^n\!L(K_i)$ , then $\pu{}^n\!L(K_i)$ is isomorphic (...) to ${}^n\!L(\pu K_i)$ . We apply our results to prove the decidability of the set of sentences true in almost all finite groups of the form L(K) where K is a finite field and L a fixed untwisted Chevalley type. (shrink)