Categorical-theoretic semantics for the relevance logic is proposed which is based on the construction of the topos of functors from a relevant algebra (considered as a preorder category endowed with the special endofunctors) in the category of sets Set. The completeness of the relevant system R of entailment is proved in respect to the semantic considered.

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 construct a 2-equivalence \(\mathfrak {CohTheory}^{op }\simeq \mathfrak {TypeSpaceFunc}\). Here \(\mathfrak {CohTheory}\) is the 2-category of positive theories and \(\mathfrak {TypeSpaceFunc}\) is the 2-category of type space functors. We give a precise definition of interpretations for positive logic, which will be the 1-cells in \(\mathfrak {CohTheory}\). The 2-cells are definable homomorphisms. The 2-equivalence restricts to a duality of categories, making precise the philosophy that a theory is ‘the same’ as the collection of its type spaces (i.e. its type space (...) functor). In characterising those functors that arise as type space functors, we find that they are specific instances of (coherent) hyperdoctrines. This connects two different schools of thought on the logical structure of a theory. The key ingredient, the Deligne completeness theorem, arises from topos theory, where positive theories have been studied under the name of coherent theories. (shrink)

Following Birkhoff and von Neumann, quantum logic has traditionally been based on the lattice of closed linear subspaces of some Hubert space, or, more generally, on the lattice of projections in a von Neumann algebra A. Unfortunately, the logical interpretation of these lattices is impaired by their nondistributivity and by various other problems. We show that a possible resolution of these difficulties, suggested by the ideas of Bohr, emerges if instead of single projections one considers elementary propositions to be families (...) of projections indexed by a partially ordered set C(A) of appropriate commutative subalgebras of A. In fact, to achieve both maximal generality and ease of use within topos theory, we assume that A is a so-called Rickart C*-algebra and that C(A) consists of all unital commutative Rickart C*-subalgebras of A. Such families of projections form a Hey ting algebra in a natural way, so that the associated propositional logic is intuitionistic: distributivity is recovered at the expense of the law of the excluded middle. Subsequently, generalizing an earlier computation for n x n matrices, we prove that the Heyting algebra thus associated to A arises as a basis for the internal Gelfand spectrum (in the sense of Banaschewski-Mulvey) of the "Bohrification" A̱ of A, which is a commutative Rickart C*-algebra in the topos of functors from C(A) to the category of sets. We explain the relationship of this construction to partial Boolean algebras and Bruns-Lakser completions. Finally, we establish a connection between probability measures on the lattice of projections on a Hubert space H and probability valuations on the internal Gelfand spectrum of A̱ for A = B(H). (shrink)

Following Birkhoff and von Neumann, quantum logic has traditionally been based on the lattice of closed linear subspaces of some Hilbert space, or, more generally, on the lattice of projections in a von Neumann algebra A. Unfortunately, the logical interpretation of these lattices is impaired by their nondistributivity and by various other problems. We show that a possible resolution of these difficulties, suggested by the ideas of Bohr, emerges if instead of single projections one considers elementary propositions to be families (...) of projections indexed by a partially ordered set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}(A)}$$\end{document} of appropriate commutative subalgebras of A. In fact, to achieve both maximal generality and ease of use within topos theory, we assume that A is a so-called Rickart C*-algebra and that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}(A)}$$\end{document} consists of all unital commutative Rickart C*-subalgebras of A. Such families of projections form a Heyting algebra in a natural way, so that the associated propositional logic is intuitionistic: distributivity is recovered at the expense of the law of the excluded middle. Subsequently, generalizing an earlier computation for n × n matrices, we prove that the Heyting algebra thus associated to A arises as a basis for the internal Gelfand spectrum (in the sense of Banaschewski–Mulvey) of the “Bohrification” \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\underline A}$$\end{document} of A, which is a commutative Rickart C*-algebra in the topos of functors from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}A}$$\end{document} to the category of sets. We explain the relationship of this construction to partial Boolean algebras and Bruns–Lakser completions. Finally, we establish a connection between probability measures on the lattice of projections on a Hilbert space H and probability valuations on the internal Gelfand spectrum of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\underline A}$$\end{document} for A = B(H). (shrink)

In this paper we adopt a category-theoretic viewpoint in order to analyze the semantics of complementarity for quantum systems. Based on the existence of a pair of adjoint functors between the topos of presheaves of the Boolean kind of structure and the category of the quantum kind of structure, we establish a twofold complementarity scheme which constitutes an instance of the concept of adjunction. It is further argued that the established scheme is inextricably connected with a realistic philosophical (...) attitude, although substantially different from the classical one. (shrink)

The overwhelming majority of the attempts in exploring the problems related to quantum logical structures and their interpretation have been based on an underlying set-theoretic syntactic language. We propose a transition in the involved syntactic language to tackle these problems from the set-theoretic to the category-theoretic mode, together with a study of the consequent semantic transition in the logical interpretation of quantum event structures. In the present work, this is realized by representing categorically the global structure of a quantum algebra (...) of events (or propositions) in terms of sheaves of local Boolean frames forming Boolean localization functors. The category of sheaves is a topos providing the possibility of applying the powerful logical classification methodology of topos theory with reference to the quantum world. In particular, we show that the topos-theoretic representation scheme of quantum event algebras by means of Boolean localization functors incorporates an object of truth values, which constitutes the appropriate tool for the definition of quantum truth-value assignments to propositions describing the behavior of quantum systems. Effectively, this scheme induces a revised realist account of truth in the quantum domain of discourse. We also include an Appendix, where we compare our topos-theoretic representation scheme of quantum event algebras with other categorial and topos-theoretic approaches. (shrink)

The Topos of Music is the upgraded and vastly deepened English extension of the seminal German Geometrie der Töne. It reflects the dramatic progress of mathematical music theory and its operationalization by information technology since the publication of Geometrie der Töne in 1990. The conceptual basis has been vastly generalized to topos-theoretic foundations, including a corresponding thoroughly geometric musical logic. The theoretical models and results now include topologies for rhythm, melody, and harmony, as well as a classification theory (...) of musical objects that comprises the topos-theoretic concept framework. Classification also implies techniques of algebraic moduli theory. The classical models of modulation and counterpoint have been extended to exotic scales and counterpoint interval dichotomies. The probably most exciting new field of research deals with musical performance and its implementation on advanced object-oriented software environments. This subject not only uses extensively the existing mathematical music theory, it also opens the language to differential equations and tools of differential geometry, such as Lie derivatives. Mathematical performance theory is the key to inverse performance theory, an advanced new research field which deals with the calculation of varieties of parameters which give rise to a determined performance. This field uses techniques of algebraic geometry and statistics, approaches which have already produced significant results in the understanding of highest-ranked human performances. The book's formal language and models are currently being used by leading researchers in Europe and Northern America and have become a foundation of music software design. This is also testified by the book's nineteen collaborators and the included CD-ROM containing software and music examples. (shrink)

The article focuses on the cartographic enactment of the topos of Ukraine as a lost homeland in contemporary German literary discourse on migration, and in particular in the body of work that conveys the voices of the “second generation” — children of the German (post-)war migration. The article analyses by way of an illustrative example Hans-Ulrich Treichel’s poem “Flight into Kyiv,” in which we find reflected the autobiographical theme of the (re)construction of the lost homeland of his father, a (...) Volyn German who fled to Germany during the Second World War to escape reprisals of the Soviet army. The main object of research are the mental maps of Ukrainian space, which largely define the way Ukraine is represented both in contemporary German social discourse and in modern German literature. A textual analysis of the poem allows us to discern how the resource of the Western European construct of Eastern Europe is instrumentalized and aesthetically arranged by German (post)migration perception to institutionalize the image of Ukraine as a “lost homeland.”. (shrink)

Terminologically, the “topos of mu” and the “predicative self” originated in the Kyoto School and are traceable to the work of its founder NISHIDA Kitarō. The full phrase was coined by NAKAMURA Yūjirō. Conceptually, the topos of mu or place of nothingness is Nishida’s development of the Buddhist notion of anatta or no self and radiating out from that locus of emptiness is a self constituted by its predicates or the things to which it is connected by an (...) existential copula. Deeply ingrained in Western languages, metaphysics, and religion is the subjective self, in both the linguistic and psychological senses of “subjective.” That Buddhism, as reworked by the Kyota School, or Daoism, or any other non-Western tradition of thought, will catch on in the West was a puerile fantasy of some members of the first generation of environmental philosophers. There is a good chance, however, that the Western worldview may evolve toward a similar conception of the self—as ecological, relational, or systems thinking becomes ever more ingrained. We in the West may come to understand that we are constituted by our social and environmental relationships, in which we are deeply embedded and on which we are utterly dependent, such that world care is the essence of self-help. (shrink)

The present work depicts the content and functions of very specific carriers of collective memory, namely the epigraphic transmissions. The rich tradition of using epigraphic objects, their durability and communicative seriousness, make them consistently attractive forms of transferring memory also in small local communities of towns, villages, parishes, schools and working commu-nities. Those seen as particularly interesting ones are epigraphic objects, whose task is to “commemorate the origins”. The use of the “topos of origins” creates an initiation moment on (...) the line of collective memory of a given local community. The observations presented in the work relate to the epigraphic resource, created in the last half century within the spatial framework of the borderland of Silesia and Lesser Poland. (shrink)

This paper focuses on the significance of the concept of noise for cognition and computation. The concept of noise was massively transformed in the twentieth century with the advent of information theory, cybernetics, and computer science, all of which provide formal accounts of information and noise centrally concerned with contingency. We show how the concept has changed from these classical formulations, through developments in mathematics (topology and topos theory), computing (interactive computing and univalent foundations), and cognitive science (predictive processing (...) and cognitive morphodynamics). Ultimately it argues for the central importance of noise not only within a topological conception of cognition and computation, but also in the transcendental-empirical torsion of image schemata and the social interactive elaboration of freedom. (shrink)

This dissertation is concerned with one of the central but most perplexing theories in Plotinus's metaphysics, namely the nature and origin of time. In contrast to those interpretations of Neoplatonism that treat time as an imperfect image and passive product of eternity, I argue for a much more subtle and multifaceted concept that makes the human observer central to Plotinus's account of how time is actualized and thus passes. His emphasis on mystical experience and the individual soul's journey toward the (...) One and the realm of eternity strengthens this account. By reading the treatise On Eternity and Time in this light, I demonstrate the importance of embodied human consciousness for the Plotinian theory of time. ;This emphasis invites comparison with the one contemporary Continental thinker who devoted countless pages in both his published work and his manuscripts to the relationship between time and consciousness. In the second part of the dissertation I argue that the Husserlian and Plotinian subjects have analogous characteristics, in accordance with their parallel roles as sources of time, and that the Husserlian notion of omnitemporality offers the best way of understanding the internal connection between Plotinus's notion of eternity and his notion of actualization through human experience. At the same time, Plotinus challenges a phenomenological account by giving the eternal ontological priority over the human realm of temporality. Nevertheless, these intriguing affinities and oppositions suggest that Plotinus's theory of time and eternity can best be understood and even expanded by uncovering its phenomenological underpinnings. (shrink)

The category-theoretic representation of quantum event structures provides a canonical setting for confronting the fundamental problem of truth valua- tion in quantum mechanics as exemplified, in particular, by Kochen-Specker’s theorem. In the present study, this is realized on the basis of the existence of a categorical adjunction between the category of sheaves of variable local Boolean frames, constituting a topos, and the category of quantum event al- gebras. We show explicitly that the latter category is equipped with an object (...) of truth values, or classifying object, which constitutes the appropriate tool for assigning truth values to propositions describing the behavior of quantum systems. Effectively, this category-theoretic representation scheme circumvents consistently the semantic ambiguity with respect to truth valuation that is in- herent in conventional quantum mechanics by inducing an objective contextual account of truth in the quantum domain of discourse. The philosophical im- plications of the resulting account are analyzed. We argue that it subscribes neither to a pragmatic instrumental nor to a relative notion of truth. Such an account essentially denies that there can be a universal context of reference or an Archimedean standpoint from which to evaluate logically the totality of facts of nature. In this light, the transcendence condition of the usual concep- tion of correspondence truth is superseded by a reflective-like transcendental reasoning of the proposed account of truth that is suitable to the quantum domain of discourse. (shrink)

It is well-established that topos theory is inherently connected with intuitionistic logic. In recent times several works appeared concerning so-called complement-toposes, which are allegedly connected to the dual to intuitionistic logic. In this paper I present this new notion, some of the motivations for it, and some of its consequences. Then, I argue that, assuming equivalence of certain two definitions of a topos, the concept of a complement-classifier is, at least in general and within the conceptual framework of (...) category theory, not appropriately defined. For this purpose, I first analyze the standard notion of a subobject classifier, show its connection with the representability of the functor Sub via the Yoneda lemma, recall some other properties of the internal structure of a topos and, based on these, I critically comment on the notion of a complement-classifier. (shrink)

The category-theoretic representation of quantum event structures provides a canonical setting for confronting the fundamental problem of truth valuation in quantum mechanics as exemplified, in particular, by Kochen-Specker’s theorem. In the present study, this is realized by representing categorically the global structure of a quantum algebra of events in terms of sheaves of local Boolean frames forming Boolean localization functors. The category of sheaves is a topos providing the possibility of applying the powerful logical classification methodology of (...) class='Hi'>topos theory with reference to the quantum world. In particular, we show that the topos-theoretic representation scheme of quantum event algebras by means of Boolean localization functors incorporates an object of truth values, which constitutes the appropriate tool for the definition of quantum truth-value assignments to propositions describing the behavior of quantum systems. Effectively, this scheme induces a contextualist account of truth in the quantum domain of discourse. The philosophical implications of the resulting account are analyzed. Such an account essentially denies that there can be a universal context of reference or an Archimedean standpoint from which to state the totality of facts of nature. (shrink)

In his philosophy of nothingness, Kitarō Nishida illuminates the matrix of transformation of the world "from the Created to the Creating" (tsukuru mono kara tsukurareta mono e) through shintai, or the body. In this matrix, shintai enters into the stage of an action-sensation continuum and emerges as the immaculate iconic tool of nothingness to create new figures as extended self. This idea of shintai has resonance with the development of postwar art in Japan. The "Space of Transparency" put forth by (...) Ufan Lee, the leader of Monoha, is the principal example. This essay investigates Nishida's notion of shintai and its influence on Lee's theory of art. (shrink)

: In his philosophy of nothingness, Kitar Nishida illuminates the matrix of transformation of the world ‘‘from the Created to the Creating’’ (tsukuru mono kara tsukurareta mono e) through shintai, or the body. In this matrix, shintai enters into the stage of an action-sensation continuum and emerges as the immaculate iconic tool of nothingness to create new figures as extended self. This idea of shintai has resonance with the development of postwar art in Japan. The ‘‘Space of Transparency’’ put forth (...) by Ufan Lee, the leader of Monoha, is the principal example. This essay investigates Nishida’s notion of shintai and its influence on Lee’s theory of art. (shrink)

In his philosophy of nothingness, Kitar Nishida illuminates the matrix of transformation of the world ''from the Created to the Creating'' through shintai, or the body. In this matrix, shintai enters into the stage of an action-sensation continuum and emerges as the immaculate iconic tool of nothingness to create new figures as extended self. This idea of shintai has resonance with the development of postwar art in Japan. The ''Space of Transparency'' put forth by Ufan Lee, the leader of Monoha, (...) is the principal example. This essay investigates Nishida's notion of shintai and its influence on Lee's theory of art. (shrink)

Zusammenfassung Der Begriff,interkultureller Konflikt' wird nicht nur in Medien und Öffentlichkeit, sondern auch in der wissenschaftlichen Diskussion zur Bezeichnung von Auseinandersetzungen verwendet, an denen Menschen mit Migrationshintergrund beteiligt sind. Den Konfliktbeteiligten werden dabei Mentalitätsunterschiede attestiert, die häufig mit Verweis auf kulturelle Differenzen erklärt werden. Der Beitrag liefert Erklärungen dafür, warum solche Auseinandersetzungen im Sport so selbstverständlich als,interkulturelle' Konflikte gekennzeichnet werden. Im Mittelpunkt der konflikt­theoretischen Analyse steht die Frage, wie das,Interkulturelle' zum Topos von Konflikten im Sport wird. Ziel ist es, (...) gängige Annahmen des Diskurses über interkulturelle Konflikte im Sport kritisch zu hinterfragen. Den theoretischen Rahmen hierfür bildet die systemtheoretische Konfliktsoziologie. (shrink)

Several Gentzen-style syntactic type calculi with product are considered. They form a hierarchy in such a way that one calculus results from another by imposing a new condition upon the sequent-forming operation. It turns out that, at some steps of this process, two different functors collapse to a single one. For the remaining stages of the hierarchy, analogues of Wajsbergs's theorem on non-mutual-definability are proved.

We prove, by using the concept of schematic interpretation, that the natural embedding from the category ISL, of intuitionistic sentential pretheories and i-congruence classes of morphisms, to the category CSL, of classical sentential pretheories and c-congruence classes of morphisms, has a left adjoint, which is related to the double negation interpretation of Gödel-Gentzen, and a right adjoint, which is related to the Law of Excluded Middle. Moreover, we prove that from the left to the right adjoint there is a pointwise (...) epimorphic natural transformation and that since the two endofunctors at CSL, obtained by adequately composing the aforementioned functors, are naturally isomorphic to the identity functor for CSL, the string of adjunctions constitutes an adjoint cylinder. On the other hand, we show that the operators of Lindenbaum-Tarski of formation of algebras from pretheories can be extended to equivalences of categories from the category CSL, respectively, ISL, to the category Bool, of Boolean algebras, respectively, Heyt, of Heyting algebras. Finally, we prove that the functor of regularization from Heyt to Bool has, in addition to its well-known right adjoint (that is, the canonical embedding of Bool into Heyt) a left adjoint, that from the left to the right adjoint there is a pointwise epimorphic natural transformation, and, finally, that such a string of adjunctions constitutes an adjoint cylinder. (shrink)

Conceptualizing the nature of reality and the way the world functions, Ekaterina Yahyaoui Krivenko analyzes the foundations of human rights law in the strict subject/object dichotomy. Seeking to dismantle this dichotomy using topo-logic, a concept developed by Japanese philosopher Nishida Kitaro, this topical book formulates ways to operationalize alternative visions of human rights practice. Subject/object dichotomy, Yahyaoui Krivenko demonstrates, emerges from and reflects a particular Western worldview through a quest for rationality and formal logic. Taking a metaphysical and epistemological perspective, (...) this book explores the alternative views of reality and logic, developed by Kitaro, to demonstrate how topo-logic can enable both a theoretical and a practical renewal of human rights and overcome the subject/object dichotomy. Examining the recent growth of social movements, decolonization and diversification of discourses about human rights, and substantive equality, the book identifies these developments in contemporary human rights as indications of a movement towards a topo-logical view beyond the subject/object dichotomy. Students and scholars of critical legal studies, legal theory and philosophy, and international human rights law will find this book to be an invigorating read. Laying ground for the possible renewal and enhancement of human rights law, it will also be a useful resource for practitioners of human rights law. (shrink)

In this article, I elucidate the significance of Heidegger’s ‘question of being’ from a topological point of view by explaining the relationship between his thought of place and language. After exploring various hermeneutic strategies of reading Heidegger’s oeuvre, I turn to Richard Capobianco’s interpretation of Heidegger and critically engage with his idea of the experience of being itself as the ‘luminous selfshowing of logos’. In doing so, I explain the later turn from ‘truth’ to ‘place’ and articulate why logos needs (...) to be conceived as the gathering site of the presencing of being. In arguing against the primacy of passive receptivity and active projection, I put forward the primacy of language as the topos and logos of being. In returning to the Capobianco-Sheehan debate, I conclude by explaining why Heidegger’s placerelated notions cannot be thought metaphorically. (shrink)

This unique book provides a self-contained conceptual and technical introduction to the theory of differential sheaves. This serves both the newcomer and the experienced researcher in undertaking a background-independent, natural and relational approach to "physical geometry". In this manner, this book is situated at the crossroads between the foundations of mathematical analysis with a view toward differential geometry and the foundations of theoretical physics with a view toward quantum mechanics and quantum gravity. The unifying thread is provided by the theory (...) of adjoint functors in category theory and the elucidation of the concepts of sheaf theory and homological algebra in relation to the description and analysis of dynamically constituted physical geometric spectrums. (shrink)

The most difficult problem that Leniewski came across in constructing his system of the foundations of mathematics was the problem of defining definitions, as he used to put it. He solved it to his satisfaction only when he had completed the formalization of his protothetic and ontology. By formalization of a deductive system one ought to understand in this context the statement, as precise and unambiguous as possible, of the conditions an expression has to satisfy if it is added to (...) the system as a new thesis. Now, some protothetical theses, and some ontological ones, included in the respective systems, happen to be definitions. In the present essay I employ Leniewski's method of terminological explanations for the purpose of formalizing ukasiewicz's system of implicational calculus of propositions, which system, without having recourse to quantification, I first extended some time ago into a functionally complete system. This I achieved by allowing for a rule of implicational definitions, which enabled me to define any propositionforming functor for any finite number of propositional arguments. (shrink)

In this document, we introduce a novel formalism for any field theory and apply it to the effective field theories of large-scale structure. The new formalism is based on functors of actions composing those theories. This new formalism predicts the actionic fields. We discuss our findings in a cosmological gravitology framework. We present these results with a cosmological inference approach and give guidelines on how we can choose the best candidate between those models with some latest understanding of model (...) selection using Bayesian inference. (shrink)

Topos Theory is an important branch of mathematical logic of interest to theoretical computer scientists, logicians and philosophers who study the foundations of mathematics, and to those working in differential geometry and continuum physics. This compendium contains material that was previously available only in specialist journals. This is likely to become the standard reference work for all those interested in the subject.

Topos Theory is a subject that stands at the junction of geometry, mathematical logic and theoretical computer science, and it derives much of its power from the interplay of ideas drawn from these different areas. Because of this, an account of topos theory which approaches the subject from one particular direction can only hope to give a partial picture; the aim of this compendium is to present as comprehensive an account as possible of all the main approaches and (...) thereby to demonstrate the overall unity of the subject. The material is organised in such a way that readers interested in following a particular line of approach may do so by starting at an appropriate point in the text. (shrink)

ArgumentGlass vessels such as flasks and test tubes play an ambiguous role in the historiography of modern laboratory research. In spite of the strong focus on the role of materiality in the last decades, the scientific glass vessel – while being symbolically omnipresent – has remained curiously neglected in regard to its materiality. The popular image ortoposof the transparent, neutral, and quasi-immaterial glass container obstructs the view of the physico-chemical functionality of this constitutive inner boundary in modern laboratory environments and (...) its material historicity. In order to understand how glass vessels were able to provide a stable epistemic containment of spatially enclosed experimental phenomena in the new laboratory ecologies emerging in the nineteenth and early twentieth century, I will focus on the history of thematerialstandardization of laboratory glassware. I will follow the rise of a new awareness for measurement errors due to the chemical agency of experimental glass vessels, then I will sketch the emergence of a whole techno-scientific infrastructure for the improvement of glass container quality in late nineteenth-century Germany. In the last part of my argument, I will return to the laboratory by looking at the implementation of this glass reform that created a newoikosfor the inner experimental milieus of modern laboratory research. (shrink)

We investigate a certain system of intuitionistic set theory from three points of view: an elementary set theory with bounded separation, a topos with distinguished inclusions, and a category of classes with a system of small maps. The three presentations are shown to be equivalent in a strong sense.