We prove the triviality of the Grothendieck ring of a Z-valued field K under slight conditions on the logical language and on K. We construct a definable bijection from the plane K 2 to itself minus a point. When we specialized to local fields with finite residue field, we construct a definable bijection from the valuation ring to itself minus a point.

We prove the triviality of the Grothendieck ring of a $\mathbb{Z}$-valued field K under slight conditions on the logical language and on K. We construct a definable bijection from the plane K$^2$ to itself minus a point. When we specialized to local fields with finite residue field, we construct a definable bijection from the valuation ring to itself minus a point.

We recall the notions of weak and strong Euler characteristics on a first order structure and make explicit the notion of a Grothendieck ring of a structure. We define partially ordered Euler characteristic and Grothendieck ring and give a characterization of structures that have non-trivial partially ordered Grothendieck ring. We give a generalization of counting functions to locally finite structures, and use the construction to show that the Grothendieck ring of the complex numbers contains as (...) a subring the ring of integer polynomials in continuum many variables. We prove the existence of a universal strong Euler characteristic on a structure. We investigate the dependence of the Grothendieck ring on the theory of the structure and give a few counter-examples. Finally, we relate some open problems and independence results in bounded arithmetic to properties of particular Grothendieck rings. (shrink)

We develop a categorical scheme of interpretation of quantum event structures from the viewpoint of Grothendieck topoi. The construction is based on the existence of an adjunctive correspondence between Boolean presheaves of event algebras and Quantum event algebras, which we construct explicitly. We show that the established adjunction can be transformed to a categorical equivalence if the base category of Boolean event algebras, defining variation, is endowed with a suitable Grothendieck topology of covering systems. The scheme leads (...) to a sheaf theoretical representation of Quantum structure in terms of variation taking place over epimorphic families of Boolean reference frames. (shrink)

We recall the notions of weak and strong Euler characteristics on a first order structure and make explicit the notion of a Grothendieck ring of a structure. We define partially ordered Euler characteristic and Grothendieck ring and give a characterization of structures that have non-trivial partially ordered Grothendieck ring. We give a generalization of counting functions to locally finite structures, and use the construction to show that the Grothendieck ring of the complex numbers contains as (...) a subring the ring of integer polynomials in continuum many variables. We prove the existence of a universal strong Euler characteristic on a structure. We investigate the dependence of the Grothendieck ring on the theory of the structure and give a few counter-examples. Finally, we relate some open problems and independence results in bounded arithmetic to properties of particular Grothendieck rings. (shrink)

A novel conceptual framework is introduced for the Complexity Levels Theory in a Categorical Ontology of Space and Time. This conceptual and formal construction is intended for ontological studies of Emergent Biosystems, Super-complex Dynamics, Evolution and Human Consciousness. A claim is defended concerning the universal representation of an item’s essence in categorical terms. As an essential example, relational structures of living organisms are well represented by applying the important categorical concept of natural transformations to biomolecular reactions and relational structures (...) that emerge from the latter in living systems. Thus, several relational theories of living systems can be represented by natural transformations of organismic, relational structures. The ascent of man and other living organisms through adaptation, is viewed in novel categorical terms, such as variable biogroupoid representations of evolving species. Such precise but flexible evolutionary concepts will allow the further development of the unifying theme of local-to-global approaches to highly complex systems in order to represent novel patterns of relations that emerge in super- and ultra-complex systems in terms of compositions of local procedures. Solutions to such local-to-global problems in highly complex systems with ‘broken symmetry’ might be possible to be reached with the help of higher homotopy theorems in algebraic topology such as the generalized van Kampen theorems (HHvKT). Categories of many-valued, Łukasiewicz-Moisil (LM) logic algebras provide useful concepts for representing the intrinsic dynamic ‘asymmetry’ of genetic networks in organismic development and evolution, as well as to derive novel results for (non-commutative) Quantum Logics. Furthermore, as recently pointed out by Baianu and Poli (Theory and applications of ontology, vol 1. Springer, Berlin, in press), LM-logic algebras may also provide the appropriate framework for future developments of the ontological theory of levels with its complex/entangled/intertwined ramifications in psychology, sociology and ecology. As shown in the preceding two papers in this issue, a paradigm shift towards non-commutative, or non-Abelian, theories of highly complex dynamics—which is presently unfolding in physics, mathematics, life and cognitive sciences—may be implemented through realizations of higher dimensional algebras in neurosciences and psychology, as well as in human genomics, bioinformatics and interactomics. (shrink)

Dynamic Predicate Logic (DPL) is a variant of Predicate Logic introduced by Groenendijk and Stokhof. One rationale behind the introduction of DPL is that it is closer to Natural Language than ordinary Predicate Logic in the way it treats scope.In this paper I develop some variants of DPL that can more easily approximate Natural Language in some further aspects. Specifically I add flexibility in the treatment of polarity and and some further flexibility in the treatment of scope.I develop a framework (...) that is intended to encourage further experimentation with alternative variants of DPL. In this framework the new meanings are, roughly,indexed sets of old meanings. The indices can be viewed as "files'' or "storage devices.''Each such file supports a separate "information stream.''The interaction of the new meanings is "programmed'' with the help of certain monoids acting on the indices. The construction of the new meanings can be viewed as an application of the Grothendieck Construction to monoids. (shrink)

We construct a sheaf-theoretic representation of quantum observables algebras over a base category equipped with a Grothendieck topology, consisting of epimorphic families of commutative observables algebras, playing the role of local arithmetics in measurement situations. This construction makes possible the adaptation of the methodology of Abstract Differential Geometry (ADG), à la Mallios, in a topos-theoretic environment, and hence, the extension of the “mechanism of differentials” in the quantum regime. The process of gluing information, within diagrams of commutative algebraic (...) localizations, generates dynamics, involving the transition from the classical to the quantum regime, formulated cohomologically in terms of a functorial quantum connection, and subsequently, detected via the associated curvature of that connection. (shrink)

We characterize the category of Sambin’s positive topologies as the result of the Grothendieck construction applied to a doctrine over the category Loc of locales. We then construct an adjunction between the category of positive topologies and that of topological spaces Top, and show that the well-known adjunction between Top and Loc factors through the constructed adjunction.

The notion of a multi-term π-institution is introduced and a criterion for the equivalence of two multi-term π-institutions in terms of their categories of theories is proved. Moreover, a counterexample that shows that this criterion is false for arbitrary π-institutions is given.

The paper investigates inaccessible set axioms and their consistency strength in constructive set theory. In ZFC inaccessible sets are of the form Vκ where κ is a strongly inaccessible cardinal and Vκ denotes the κth level of the von Neumann hierarchy. Inaccessible sets figure prominently in category theory as Grothendieck universes and are related to universes in type theory. The objective of this paper is to show that the consistency strength of inaccessible set axioms heavily depend on the context (...) in which they are embedded. The context here will be the theory CZF− of constructive Zermelo–Fraenkel set theory but without -Induction . Let INAC be the statement that for every set there is an inaccessible set containing it. CZF−+INAC is a mathematically rich theory in which one can easily formalize Bishop style constructive mathematics and a great deal of category theory. CZF−+INAC also has a realizability interpretation in type theory which gives its theorems a direct computational meaning. The main result presented here is that the proof theoretic ordinal of CZF−+INAC is a small ordinal known as the Feferman–Schütte ordinal Γ0. (shrink)

We characterize the category of Sambin’s positive topologies as the result of the Grothendieck construction applied to a doctrine over the category Loc of locales. We then construct an adjunction between the category of positive topologies and that of topological spaces Top, and show that the well-known adjunction between Top and Loc factors through the constructed adjunction.

We give an explicit construction of the dependent product in an elementary topos, and a site‐theoretic description for it in the case of a Grothendieck topos. Along the way, we obtain a number of results of independent interest, including an expression for the operation of universal quantification on subobjects in terms of finite limits and power objects.

This book explores and articulates the concepts of the continuous and the infinitesimal from two points of view: the philosophical and the mathematical. The first section covers the history of these ideas in philosophy. Chapter one, entitled ‘The continuous and the discrete in Ancient Greece, the Orient and the European Middle Ages,’ reviews the work of Plato, Aristotle, Epicurus, and other Ancient Greeks; the elements of early Chinese, Indian and Islamic thought; and early Europeans including Henry of Harclay, Nicholas of (...) Autrecourt, Duns Scotus, William of Ockham, Thomas Bradwardine and Nicolas Oreme. The second chapter of the book covers European thinkers of the sixteenth and seventeenth centuries: Galileo, Newton, Leibniz, Descartes, Arnauld, Fermat, and more. Chapter three, 'The age of continuity,’ discusses eighteenth century mathematicians including Euler and Carnot, and philosophers, among them Hume, Kant and Hegel. Examining the nineteenth and early twentieth centuries, the fourth chapter describes the reduction of the continuous to the discrete, citing the contributions of Bolzano, Cauchy and Reimann. Part one of the book concludes with a chapter on divergent conceptions of the continuum, with the work of nineteenth and early twentieth century philosophers and mathematicians, including Veronese, Poincaré, Brouwer, and Weyl. Part two of this book covers contemporary mathematics, discussing topology and manifolds, categories, and functors, Grothendieck topologies, sheaves, and elementary topoi. Among the theories presented in detail are non-standard analysis, constructive and intuitionist analysis, and smooth infinitesimal analysis/synthetic differential geometry. No other book so thoroughly covers the history and development of the concepts of the continuous and the infinitesimal. (shrink)

In this paper we bring together results from a series of previous papers to prove the constructive version of the Gelfand duality theorem in any Grothendieck topos , obtaining a dual equivalence between the category of commutative C*-algebras and the category of compact, completely regular locales in the topos.

We present two of the three major steps in the construction of motivic integration, that is, a homomorphism between Grothendieck semigroups that are associated with a first-order theory of algebraically closed valued fields, in the fundamental work of Hrushovski and Kazhdan [8]. We limit our attention to a simple major subclass of V-minimal theories of the form ACV FS, that is, the theory of algebraically closed valued fields of pure characteristic 0 expanded by a -generated substructure S in (...) the language . The main advantage of this subclass is the presence of syntax. It enables us to simplify the arguments with many different technical details while following the major steps of the Hrushovski–Kazhdan theory. (shrink)

We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by Lawvere-Tierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming the existing topos-theoretic results.

We present a conjecture in Diophantine geometry concerning the construction of line bundles over smooth projective varieties over ${\overline {\mathbb {Q}}}$. This conjecture, closely related to the Grothendieck period conjecture for cycles of codimension $1$, is also motivated by classical algebraization results in analytic and formal geometry and in transcendence theory. Its formulation involves the consideration of $D$-group schemes attached to abelian schemes over algebraic curves over ${\overline {\mathbb {Q}}}$. We also derive the Grothendieck period conjecture for (...) cycles of codimension $1$ in abelian varieties over ${\overline {\mathbb {Q}}}$ from a classical transcendence theorem à la Schneider–Lang. (shrink)

We use the language of categorical logic to construct generic saturated models of intuitionistic theories. Our main technique is the thorough study of the filter construction on categories with finite limits, which is the completion of subobject lattices under filtered meets. When restricted to coherent or Heyting categories, classifying categories of intuitionistic first-order theories, the resulting categories are filtered meet coherent categories, coherent categories with complete subobject lattices such that both finite disjunctions and existential quantification distribute over filtered meets. (...) Such categories naturally embed into Grothendieck toposes which then contain saturated models of the theory we started with. (shrink)

Alexandre Grothendieck foi um dos maiores matemáticos do século 20 e um dos mais atípicos. Nascido na Alemanha a um pai anarquista de origem russa, sua infância foi marcada pela militância política dos seus pais, assim passando por revoluções, guerras e sobrevivência. Descoberto por sua precocidade matemática por Henri Cartan, Grothendieck fez seu doutorado sob orientação de Laurent Schwartz e Jean Dieudonné. As principais contribuições dele são na área da topologia e na geometria algébrica, assim como na teoria (...) das categorias. No final dos anos de 1960, ele se dedicou à militância política e ecológica, organizando a revista Survivre durante três anos. Em 1986, publicou um manuscrito autobiográfico de 1000 páginas, Récoltes et semailles, em que ele descreve sua experiência e sua prática da matemática, assim suas contribuições à comunidade matemática francesa. Pouco comentado na filosofia, as implicações dos seus descobrimentos fora mais recentemente discutidas por Alain Badiou na sua "fenômeno-lógica", em Logiques des mondes e Arkady Plonitsky, Mathgematics, Science and postclassical Theory, pesquisa trata da semelhança entre os aspectos formais da filosofia de Gilles Deleuze e da topologia de Grothendieck. (shrink)

In this paper we define a notion of relativization for higher order logic. We then show that there is a higher order theory of Grothendieck topoi such that all Grothendieck topoi relativizes to all models of set theory with choice.

No mathematician did more to change mathematics in the second half of the twentieth century than Alexandre Grothendieck. This would have been true even if he had been a quiet figure with a liking for playing the piano and walking in the hills but, as this book makes very clear, he was far from that, and his character and his way of working enhanced his impact. Above all, there was his abrupt departure from the world of mathematics in 1970 (...) and his occasional interventions in it since.This review was submitted a few days before Grothendieck's death in November 2014—Editor.As a teenager, Grothendieck had lived with his mother protected from the Nazis by courageous Huguenots in the village of Le Chambon-sur-Lignon. Immediately after the war, he went to the University of Montpellier where he studied mathematics and soon made his way to Nancy and was introduced to Dieudonné and Schwartz, who were members of Bourbaki. His first original work was in the theory of Banach spaces where, in the wo .. (shrink)

We shall address from a conceptual perspective the duality between algebra and geometry in the framework of the refoundation of algebraic geometry associated to Grothendieck’s theory of schemes. To do so, we shall revisit scheme theory from the standpoint provided by the problem of recovering a mathematical structure A from its representations \ into other similar structures B. This vantage point will allow us to analyze the relationship between the algebra-geometry duality and the structure-semiotics duality. Whereas in classical algebraic (...) geometry a certain kind of rings can be recovered by considering their representations with respect to a unique codomain B, Grothendieck’s theory of schemes permits to reconstruct general rings by considering representations with respect to a category of codomains. The strategy to reconstruct the object from its representations remains the same in both frameworks: the elements of the ring A can be realized—by means of what we shall generally call Gelfand transform—as quantities on a topological space that parameterizes the relevant representations of A. As we shall argue, important dualities in different areas of mathematics can be understood as particular cases of this general pattern. In the wake of Majid’s analysis of the Pontryagin duality, we shall propose a Kantian-oriented interpretation of this pattern. We shall use this conceptual framework to argue that Grothendieck’s notion of functor of points can be understood as a “relativization of the a priori” that generalizes the relativization already conveyed by the notion of domain extension to more general variations of the corresponding domains. (shrink)

The large-structure tools of cohomology including toposes and derived categories stay close to arithmetic in practice, yet published foundations for them go beyond ZFC in logical strength. We reduce the gap by founding all the theorems of Grothendieck’s SGA, plus derived categories, at the level of Finite-Order Arithmetic, far below ZFC. This is the weakest possible foundation for the large-structure tools because one elementary topos of sets with infinity is already this strong.

This chapter first distinguishes between originality and creativity. True originality is rare, whether in art, science, or LEGO, because to be truly original means to have done something that no one has ever done before, and that no one could have anticipated. Most LEGO creations will not meet that condition, for with the exception of serious hobbyists who undertake massive builds, most players who make original creations are making creations that are commonplace. Painting or remolding or placing stickers on the (...) bricks counts against the spirit of LEGO creation. The chapter then discusses The LEGO Movie embodies the paradox that presents three conflicting models of creative LEGO play, illustrated by the Master Builders, Finn's father, and Emmet. The movie's creators were deliberately playing around with conflicting popular conceptions of creativity: creativity as madness, creativity as thinking outside the box, and creativity as vision. (shrink)

Dans sa thèse complémentaire intitulée « Essai sur l’unité des sciences mathématiques dans leur développement actuel » Albert Lautman analysa la question de l’unité des mathématiques en considérant différentes paires antithétiques de concepts mathématiques, notamment le continu et le discret. Dans le cadre de sa refonte de la géométrie algébrique abstraite, le mathématicien français Alexandre Grothendieck considéra également l’opposition traditionnelle du continu et du discret selon un cadre conceptuel fort similaire à celui de Lautman. En comparaison, l’introduction du concept (...) de topos lui permit de donner une réponse strictement mathématique et parfaitement claire à cette question.Albert Lautman’s complementary thesis, entitled “Essai sur l’unité des sciences mathématiques dans leur développement actuel”, analysed the unity of mathematics through various antithetical pairs of mathematical concepts such as the continuous and the discrete. As part of his renewal of abstract algebraic geometry, French mathematician Alexandre Grothendieck also considered the traditional opposition between the continuous and the discrete and adopted a conceptual framework very similar to that of Lautman. However, the topos concept allowed Grothendieck to come up with a purely mathematical and, in comparison, much clearer solution. (shrink)

The literature on the construction of places of worship has predominantly shown difficulties, rejection and disharmony among religious communities. This study aims to describe and analyzed the success story of the construction of the Santa Monica church in Tangerang. This is a qualitative study conducted over a month-long period using primary and secondary data. Primary data were obtained through observation and interviews, while secondary data were obtained through formal and informal policy reviews available online. The informants involved in (...) this study amounted to eight people consisting of church leaders and administrators, local leaders, religious leaders, religious organizations and the Ministry of Religious Affairs. Data were analyzed using content analysis methods. The results of the study show that the ability of church leaders and administrators to build good social and cultural relations with the community was a key element in their success in obtaining permission to build the church. This was achieved through effective communication, dialogue and cooperation between the Catholic community, leaders and the predominantly Islamic local community. Active involvement in daily activities had a positive impact, fostering mutual trust and peaceful coexistence, ultimately resulting in the local community’s support and approval for the establishment of the place of worship. Thus, the church construction committee was able to meet all administrative requirements as stipulated by government regulations. This study recommends the importance of good social and cultural relations as the fundamental basis and determinant of success for religious communities seeking to construct places of worship.Contribution: This article contributes to the goal of HTS Theological Studies in promoting the transdisciplinary aspect of religious studies in the international theological arena, particularly in the study of religious harmony among communities in Indonesia. (shrink)

Niche Construction Theory has been gaining acceptance as an explanatory framework for processes in biological and human evolution. Human cultural niche construction, in particular, is suggested as a basis for understanding many phenomena that involve human genetic and cultural evolution. Herein I assess the ability of the cultural niche construction framework to meet this explanatory role by looking into several NCT-inspired accounts that have been offered for two important episodes of human evolution, and by examining the contribution (...) of NCT to the elucidation of two “primary examples” mentioned often in the NCT literature. The result, I claim, is rather disappointing: While NCT may serve as a descriptive framework for these phenomena, it cannot be said to explain them in any substantive sense. Especially disturbing is NCT’s failure to account for differing developments in very similar situations, and to facilitate evaluation and discrimination between divergent and contradictory causal accounts of particular phenomena. I argue that these problems are inherent, and they render NCT unsuitable to serve as an explanatory framework for human phenomena. NCT’s value, at least as related to human phenomena, is therefore descriptive and heuristic rather than explanatory. In conclusion, I discuss and reject comparisons made between NCT and the theory of natural selection, and examine several potential sources of NCT’s explanatory weakness. (shrink)

In recent years sociological research on social movements has identified emotional dynamics in all the basic processes and phases of protest, and we are only beginning to understand their causal impacts. These include the solidarities of groups, motivations for action, the role of morality in political action, and the gendered division of labor in social movements. Anger turns out to be at the core of many of these causal mechanisms.

Working in a fixed Grothendieck topos Sh(C, J C ) we generalize \({\mathcal{L}_{{\infty},\omega}}\) to allow our languages and formulas to make explicit reference to Sh(C, J C ). We likewise generalize the notion of model. We then show how to encode these generalized structures by models of a related sentence of \({\mathcal{L}_{{\infty},\omega}}\) in the category of sets and functions. Using this encoding we prove analogs of several results concerning \({\mathcal{L}_{{\infty},\omega}}\) , such as the downward Löwenheim–Skolem theorem, the completeness theorem (...) and Barwise compactness. (shrink)

Inspired by Rudolf Carnap's Der Logische Aufbau Der Welt, David J. Chalmers argues that the world can be constructed from a few basic elements. He develops a scrutability thesis saying that all truths about the world can be derived from basic truths and ideal reasoning. This thesis leads to many philosophical consequences: a broadly Fregean approach to meaning, an internalist approach to the contents of thought, and a reply to W. V. Quine's arguments against the analytic and the a priori. (...) Chalmers also uses scrutability to analyze the unity of science, to defend a conceptual approach to metaphysics, and to mount a structuralist response to skepticism. Based on the 2010 John Locke lectures, Constructing the World opens up debate on central philosophical issues involving language, consciousness, knowledge, and reality. This major work by a leading philosopher will appeal to philosophers in all areas. This entry contains uncorrected proofs of front matter, chapter 1, and first excursus. (shrink)

Pierre-Henri Castel provides a short but richly argued precis of his recently published two-volume 1,000-page masterwork on the history of obsessive-compulsive disorder. Having not read the as-yet-untranslated books, I write this commentary from Plato’s cave, trying to infer the reality of Castel’s analysis from expository shadows. I am unlikely to be more successful than Plato’s poor troglodytes, so I apologize ahead of time for any misunderstandings. Moreover, I cannot assess Castel’s detailed evidential case for his substantive theses.1 I thus focus (...) on some key philosophical issues that impinge on an area of my concern, the concept of mental disorder. Castel is a rare breed of French.. (shrink)

In The Construction of Social Reality, John Searle argues that there are two kinds of facts--some that are independent of human observers, and some that require..

The aim of this paper is to bring recent work on metaphysical grounding to bear on the phenomenon of social construction. It is argued that grounding can be used to analyze social construction and that the grounding framework is helpful for articulating various claims and commitments of social constructionists, especially about social identities, e.g., gender and race. The paper also responds to a number of objections that have been leveled against the application of grounding to social construction (...) from Elizabeth Barnes, Mari Mikkola, and Jessica Wilson. (shrink)

In this paper, a family of paraconsistent propositional logics with constructive negation, constructive implication, and constructive co-implication is introduced. Although some fragments of these logics are known from the literature and although these logics emerge quite naturally, it seems that none of them has been considered so far. A relational possible worlds semantics as well as sound and complete display sequent calculi for the logics under consideration are presented.