Here we work in an arbitrary o-minimal expansion of a divisible ordered abelian group. We say that a definable ring is definably semiprime if squares of nontrivial two-sided ideals definable in the expansion are nontrivial. We prove a definable version of Wedderburn–Artin theorem and give a characterization of definably semiprime rings.

Schur’s Lemma says that the endomorphism ring of a simple left R-module is a division ring. It plays a fundamental role to prove classical ring structure theorems like the Jacobson Density Theorem and the Wedderburn–Artin Theorem. We first define the endomorphism ring of simple left R-modules by their Π10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi ^{0}_{1}$$\end{document} subsets and show that Schur’s Lemma is provable in RCA0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm (...) RCA_{0}$$\end{document}. A ring R is left primitive if there is a faithful simple left R-module and left semisimple if the left regular module RR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{R}R$$\end{document} is semisimple. The Jacobson Density Theorem and the Wedderburn-Artin Theorem characterize left primitive ring and left semisimple ring, respectively. We then study such theorems from the standpoint of reverse mathematics. (shrink)

This paper studies the structure of semisimple rings using techniques of reverse mathematics, where a ring is left semisimple if the left regular module is a finite direct sum of simple submodules. The structure theorem of left semisimple rings, also called Wedderburn-Artin Theorem, is a famous theorem in noncommutative algebra, says that a ring is left semisimple if and only if it is isomorphic to a finite direct product of matrix rings over division rings. We provide a (...) proof for the theorem in $$\mathrm RCA_{0}$$, showing the structure theorem for computable semisimple rings. The decomposition of semisimple rings as finite direct products of matrix rings over division rings is unique. Based on an effective proof of the Jordan-Hölder Theorem for modules with composition series, we also provide an effective proof for the uniqueness of the matrix decomposition of semisimple rings in $$\mathrm RCA_{0}$$. (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)

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)

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)

Profinite groups are Galois groups. The effective study of infinite Galois groups was initiated by Metakides and Nerode [8] and further developed by LaRoche [5]. In this paper we study profinite groups without considering Galois extensions of fields. The Artin method of representing a finite group as a Galois group has been generalized by Waterhouse [14] to profinite groups. Thus, there is no loss of relevance in our approach.The fundamental notions of a co-r.e. profinite group, recursively profinite group, and the (...) degree of a co-r.e. profinite group are defined in §1. In this section we prove that every co-r.e. profinite group can be effectively represented as an inverse limit of finite groups. The degree invariant is shown to behave very well with respect to open subgroups and quotients. The work done in this section is basic to the rest of the paper.The commutator subgroup, the Frattini subgroup, thep-Sylow subgroups, and the center of a profinite group are essential in the study of profinite groups. It is only natural to ask if these subgroups are effective. The following question exemplifies our approach to this problem: Is the center a co-r.e. profinite group? Theorem 2 provides a general method for answering this type of question negatively. Examples 3,4 and 5 are all applications of this theorem. (shrink)

Arguing against the dominant developmental theories (e.g., Piaget, 1945; Vygotsky, 1978) stating that pretend play is limited to early childhood, we illustrate that pretend play is an adaptive human activity of adulthood as well as childhood. We advance this argument on three levels. First, we offer an analysis of why the discipline of developmental psychology in the Western world considered play only as an activity of childhood by neglecting to explore whether or how pretend play exists during adulthood. Second, we (...) discuss the similarities between adult improvisational theater and children’s pretend play in illustrating our thesis that pretend continues to exist during adulthood. In this discussion, we focus on similarities in the definitions, psychological origins, social functions, and developmental consequences of pretend play and adult improvisation. Finally, we end the article with educational implications of conceptualizing pretend play as a life-span activity and offer directions for future research. (shrink)

This essay distinguishes between metaphysical and epistemological conceptions of analyticity. The former is the idea of a sentence that is ‘true purely in virtue of its meaning’ while the latter is the idea of a sentence that ‘can be justifiably believed merely on the basis of understanding its meaning’. It further argues that, while Quine may have been right to reject the metaphysical notion, the epistemological notion can be defended from his critique and put to work explaining a priori justification. (...) Along the way, a number of further distinctions relevant to the theory of analyticity and the theory of apriority are made and their significance is explained. (shrink)

The God of Jesus -- The nature of God: an unanswerable question? -- The nature of God in Christian tradition -- And our God?

A stellar line-up of leading philosophers from around the world offer new treatments of a topic which has long been central to philosophical debate, and in ...

This chapter aims to provide materials with which to substantiate the claim that, under the appropriate circumstances, the notion of analyticity can help explain how one might have a priori knowledge even in the strong sense. It argues that Implicit Definition, properly understood, is completely independent of any form of irrealism about logic. The chapter defends the thesis of Implicit Definition against Quine's criticisms, and examines the sort of account of the apriority of logic that this doctrine is able to (...) provide. The chapter shows that, against the background of a rejection of indeterminacy, its insolubility cannot be conceded. It also argues that neither a non‐factualism about Frege‐analyticity, nor an error thesis about it, can plausibly fall short of an outright rejection of meaning itself. The chapter shows how the doctrine that appears to offer the most promising account of how we grasp the meanings of the logical constants. (shrink)

The book records a series of philosophical exchanges between its authors, amounting to a debate extended over more than fifteen years. Its subject matter is the nature and scope of reason. A central case at issue is basic logical knowledge, and the justification for basic deductive inferences, but the arguments range far more widely, at stake the distinctions between analytic and synthetic, and between a priori and a posteriori. The discussion naturally involves problems about the conditions for linguistic understanding and (...) competence, and what it might be to grasp a concept or to have an intuition. Since reason is central to philosophical method, there are associated implications for how philosophy itself works, or should work. In particular, the discussion raises fundamental concerns about how to approach epistemology. -/- . (shrink)

Research on the metaphorical mapping of valenced concepts onto space indicates that positive, neutral, and negative concepts are mapped onto upward, midward, and downward locations, respectively. More recently, this type of research has been tested for the very first time in 3D physical space. The findings corroborate the mapping of valenced concepts onto the vertical space as described above but further show that positive and negative concepts are placed close to and away from the body; neutral concepts are placed midway. (...) The current study aimed at investigating whether valenced perceptual stimuli are positioned onto 3D space akin to the way valenced concepts are positioned. By using a unique device known as the cognition cube, participants placed visual, auditory, tactile and olfactory stimuli on 3D space. The results mimicked the placing of valenced concepts onto 3D space; i.e., positive percepts were placed in upward and close-to-the-body locations and negative percepts were placed in downward and away-from-the-body locations; neutral percepts were placed midway. These pattern of results was more pronounced in the case of visual stimuli, followed by auditory, tactile, and olfactory stimuli. Significance Statement Just recently, a unique device called “the cognition cube” (CC) enabled to find that positive words are mapped onto upward and close-to-the-body locations and negative words are mapped onto downward and away-from-the-body locations; neutral words are placed midway. This way of placing words in relation to the body is consistent with an approach-avoidance effect such that “good” and “bad” things are kept close to and away from one’s body. We demonstrate for the very first time that this same pattern emerges when visual, auditory, tactile, and olfactory perceptual stimuli are placed on 3D physical space. We believe these results are significant in that the CC can be used as a new tool to diagnose emotion-related disorders. (shrink)

Although taste and smell seem hard to imagine, some people nevertheless report vivid imagery in these sensory modalities. We investigate whether experts are better able to imagine smells and tastes because they have learned the ability, or whether they are better imaginers in the first place, and so become experts. To test this, we first compared a group of wine experts to yoked novices using a battery of questionnaires. We show for the first time that experts report greater vividness of (...) wine imagery, with no difference in vividness across sensory modalities. In contrast, novices had more vivid color imagery than taste or odor imagery for wines. Experts and novices did not differ on other vividness of imagery measures, suggesting a domain‐specific effect of expertise. Critically, in a second study, we followed a group of students commencing a wine course and a group of matched control participants. Students and controls did not differ before the course, but after the wine course students reported more vivid wine imagery. We provide evidence that expertise improves imagery, exemplifying the extent of plasticity of cognition underlying the chemical senses. (shrink)

The valence–space metaphor research area investigates the metaphorical mapping of valenced concepts onto space. Research findings from this area indicate that positive, neutral, and negative concepts are associated with upward, midward, and downward locations, respectively, in the vertical plane. The same research area has also indicated that such concepts seem to have no preferential location on the horizontal plane. The approach–avoidance effect consists in decreasing the distance between positive stimuli and the body and increasing the distance between negative stimuli and (...) the body. Thus, the valence–space metaphor accounts for the mapping of valenced concepts onto the vertical and horizontal planes, and the approach–avoidance effect accounts for the mapping of valenced concepts onto the “depth” plane. By using a cube conceived for the study of allocation of valenced concepts onto 3D space, we show in three studies that positive concepts are placed in upward locations and near the participants’ body, negative concepts are placed in downward locations and far from the participants’ body, and neutral concepts are placed in between these concepts in both planes. (shrink)

A search is under way for a theory that can accommodate our intuitions in population axiology. The object of this search has proved elusive. This is not surprising since, as we shall see, any welfarist axiology that satisfies three reasonable conditions implies at least one of three counter-intuitive conclusions. I shall start by pointing out the failures in three recent attempts to construct an acceptable population axiology. I shall then present an impossibility theorem and conclude with a short discussion (...) of how it might be extended to pluralist axiologies, that is, axiologies that take more values than welfare into account. (shrink)

A collaboration between distinguished physicists and philosophers of physics, this important anthology surveys the deep implications of Bell's nonlocality theorem.

A brief, non-technical introduction to technical and philosophical aspects of Frege's philosophy of arithmetic. The exposition focuses on Frege's Theorem, which states that the axioms of arithmetic are provable, in second-order logic, from a single non-logical axiom, "Hume's Principle", which itself is: The number of Fs is the same as the number of Gs if, and only if, the Fs and Gs are in one-one correspondence.

Kenneth Arrow’s “impossibility” theorem—or “general possibility” theorem, as he called it—answers a very basic question in the theory of collective decision-making. Say there are some alternatives to choose among. They could be policies, public projects, candidates in an election, distributions of income and labour requirements among the members of a society, or just about anything else. There are some people whose preferences will inform this choice, and the question is: which procedures are there for deriving, from what is (...) known or can be found out about their preferences, a collective or “social” ordering of the alternatives from better to worse? The answer is startling. Arrow’s theorem says there are no such procedures whatsoever—none, anyway, that satisfy certain apparently quite reasonable assumptions concerning the autonomy of the people and the rationality of their preferences. The technical framework in which Arrow gave the question of social orderings a precise sense and its rigorous answer is now widely used for studying problems in welfare economics. The impossibility theorem itself set much of the agenda for contemporary social choice theory. Arrow accomplished this while still a graduate student. In 1972, he received the Nobel Prize in economics for his contributions. (shrink)

Often, when one faces a choice between alternative actions, there are reasons both for and against each alternative. On one way of understanding these words, what one “ought to do all things considered (ATC)” is determined by the totality of these reasons. So, these reasons can somehow be “combined” or “aggregated” to yield an ATC verdict on these alternatives. First, various assumptions about this sort of aggregation of reasons are articulated. Then it is shown that these assumptions allow for the (...) proof of a “Reasons Aggregation Theorem” – parallel to John Harsanyi’s 1955 “Social Aggregation Theorem”. All reasons for action are grounded in reason-providing values; and, for every such reason-providing value, there is in principle a way of measuring how strongly this value counts against each alternative. The theorem tells us that the appropriate measure of how much reason ATC there is against each alternative is simply a weighted sum of the strengths with which these values count against that alternative; the agent ought not ATC to perform an action iff there is ATC more reason against it than against some available alternative. (shrink)

This article considers social choice theory without the Pareto principle. We revisit the trade-off theorem developed by Campbell and Kelly (Econometrica 61:1355–1365, 1993) and generalize their result. By introducing an alternative measure of decisive structure, a dominance relation, we show that if a social welfare function dominates another social welfare function, then the number of pairs of alternatives which social ranking is independently of individual preferences under the former is not more than that under the latter. Moreover, we offer (...) two applications of our main result. (shrink)

I compare deterministic and stochastic hidden variable models of the Bell experiment, exphasising philosophical distinctions between the various ways of combining conditionals and probabilities. I make four main claims. (1) Under natural assumptions, locality as it occurs in these models is equivalent to causal independence, as analysed (in the spirit of Lewis) in terms of probabilities and conditionals. (2) Stochastic models are indeed more general than deterministic ones. (3) For factorizable stochastic models, relativity's lack of superluminal causation does not favour (...) locality over completeness. (4) If we prohibit all superluminal causation, then the violation of the Bell inequality teaches us a lesson, besides quantum mechanics' familiar ones that quantities can lack precise values and that pairs of quantities can lack joint probabilities: namely, some pairs of events are not screened off by their common past. (shrink)

This paper provides a philosophical analysis of the ongoing controversy surrounding R.A. Fisher's famous ‘fundamental theorem’ of natural selection. The difference between the ‘traditional’ and ‘modern’ interpretations of the theorem is explained. I argue that proponents of the modern interpretation have captured Fisher's intended meaning correctly and shown that the theorem is mathematically correct, pace the traditional consensus. However, whether the theorem has any real biological significance remains an unresolved issue. I argue that the answer depends (...) on whether we accept Fisher's non-standard notion of environmental change, on which the theorem rests; arguments for and against this notion are explored. I suggest that there is a close link between Fisher's fundamental theorem and the modern ‘gene's eye’ view of evolution. Introduction What Does the Fundamental Theorem Say? Key Concepts Explained Alleged Significance of the FTNS Traditional versus Modern Interpretations of the FTNS The Modern Interpretation Illustrated Fisher's Concept of ‘Environmental Change’ Causality and the Modern Interpretation The Significance of the FTNS Re-considered Appendix CiteULike Connotea Del.icio.us What's this? (shrink)

Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss the algebraization of substructural logics over the full Lambek calculus and their connections to residuated lattices, and establish a weak form of the deduction theorem that is known as parametrized local deduction theorem. Finally, we study certain interpolation properties and explain how they imply the amalgamation property for certain varieties of residuated lattices.

I note that the logics of the relevant group most closely tied to the research programme in paraconsistency are those without the contraction postulate(A.AB).AB and its close relatives. As a move towards gaining control of the contraction-free systems I show that they are prime (that wheneverA B is a theorem so is eitherA orB). The proof is an extension of the metavaluational techniques standardly used for analogous results about intuitionist logic or the relevant positive logics.

Attempts to articulate the real meaning or ultimate significance of a famous theorem comprise a major vein of philosophical writing about mathematics. The subfield of mathematical logic has supplie...

In this paper, an incompleteness theorem for modal extensions of relevant logics is proved. The proof uses elementary methods and builds upon the work of Fuhrmann.

The problem of multiple-computations discovered by Hilary Putnam presents a deep difficulty for functionalism (of all sorts, computational and causal). We describe in out- line why Putnam’s result, and likewise the more restricted result we call the Multiple- Computations Theorem, are in fact theorems of statistical mechanics. We show why the mere interaction of a computing system with its environment cannot single out a computation as the preferred one amongst the many computations implemented by the system. We explain why (...) nonreductive approaches to solving the multiple- computations problem, and in particular why computational externalism, are dualistic in the sense that they imply that nonphysical facts in the environment of a computing system single out the computation. We discuss certain attempts to dissolve Putnam’s unrestricted result by appealing to systems with certain kinds of input and output states as a special case of computational externalism, and show why this approach is not workable without collapsing to behaviorism. We conclude with some remarks about the nonphysical nature of mainstream approaches to both statistical mechanics and the quantum theory of measurement with respect to the singling out of partitions and observables. (shrink)

The aim of this paper is to argue about the impossibility of constructing a complete formal theory or a complete Turing machines' algorithm that represent the human capacity of recognizing mathematical truths. More specifically, based on a direct argument from Gödel's First Incompleteness Theorem, we discuss the impossibility of constructing a complete formal theory or a complete Turing machines' algorithm to the human capacity of recognition of first-order arithmetical truths and so of mathematical truths in general.

We show that \\) cannot be proved with one typical application of \\) in an intuitionistic extension of \ to higher types, but that this does not remain true when the law of the excluded middle is added. The argument uses Kohlenbach’s axiomatization of higher order reverse mathematics, results related to modified reducibility, and a formalization of Weihrauch reducibility.

According to a widespread view, the Bell theorem establishes the untenability of so-called ‘local realism’. On the basis of this view, recent proposals by Leggett, Zeilinger and others have been developed according to which it can be proved that even some non-local realistic theories have to be ruled out. As a consequence, within this view the Bell theorem allows one to establish that no reasonable form of realism, be it local or non-local, can be made compatible with the (...) (experimentally tested) predictions of quantum mechanics. In the present paper it is argued that the Bell theorem has demonstrably nothing to do with the ‘realism’ as defined by these authors and that, as a consequence, their conclusions about the foundational significance of the Bell theorem are unjustified. (shrink)