This paper formulates a notion of independence of subobjects of an object in a general (i.e. not necessarily concrete) category. Subobject independence is the categorial generalization of what is known as subsystem independence in the context of algebraic relativistic quantum field theory. The content of subobject independence formulated in this paper is morphism co-possibility: two subobjects of an object will be defined to be independent if any two morphisms on the two subobjects of an object are jointly implementable (...) by a single morphism on the larger object. The paper investigates features of subobject independence in general, and subobject independence in the category of C∗ - algebras with respect to operations (completely positive unit preserving linear maps on C∗ - algebras)as morphisms is suggested as a natural subsystem independence axiom to express relativistic locality of the covariant functor in the categorial approach to quantum field theory. (shrink)

Category theory doesn't support Mathematical Structuralism but suggests a new philosophical view on mathematics, which differs both from Structuralism and from traditional Substantialism about mathematical objects. While Structuralism implies thinking of mathematical objects up to isomorphism the new categorical view implies thinking up to general morphism.

Focusing on the tendency of terrorist organizations to explosive attack, this article applied the institutional theory as the basis to explain the inherent logic of attack type similarity from the perspective of mimetic, coercive, and normative isomorphism. Subsequently, the study conducted an empirical analysis of the data onto 1825 terrorist organizations recorded in the Global Terrorism Database with the logistic regression method. The results show that: (1) Terrorist organizations will learn from pre-existing terrorist organizations' experiences, and mimetic isomorphism will (...) promote explosive tendency; (2) Due to the normative isomorphism effect, terrorist groups' tendency to explosive attacks is weakened by their increased duration; (3) If terrorist organizations are hostile to a strong government, coercive isomorphism positively moderates the negative effects of increasing duration. The study suggests that counter-terrorism approaches such as destroying the learnable experience of attacks, addressing the root causes of terrorism, and maintaining a strong government may be helpful in stopping increasing terrorist activities, which is essential for reducing terrorist organizations' vivosphere, blocking the inter-flow and imitation between terrorist organizations, and ultimately interrupting the terrorist propagation chain. (shrink)

is a presentation of mathematics in terms of the fundamental concepts of transformation, and composition of transformations. While the importance of these concepts had long been recognized in algebra (for example, by Galois through the idea of a group of permutations) and in geometry (for example, by Klein in his Erlanger Programm), the truly universal role they play in mathematics did not really begin to be appreciated until the rise of abstract algebra in the 1930s. In abstract algebra the idea (...) of transformation of structure (homomorphism) was central from the beginning, and it soon became apparent to algebraists that its most important concepts and constructions were in fact formulable in terms of that idea alone. Thus emerged the view that the essence of a mathematical structure is to be sought not in its internal constitution, but rather in the nature of its relationships with other structures of the same kind, as manifested through the network of transformations. This idea has achieved its fullest expression in category theory, an axiomatic framework within which the notions of transformation (as morphism or arrow) and composition (and also structure, as object) are fundamental, that is, are not defined in terms of anything else. (shrink)

Starting from a generalization of the standard axioms for a monoid we present a stepwise development of various, mutually equivalent foundational axiom systems for category theory. Our axiom sets have been formalized in the Isabelle/HOL interactive proof assistant, and this formalization utilizes a semantically correct embedding of free logic in classical higher-order logic. The modeling and formal analysis of our axiom sets has been significantly supported by series of experiments with automated reasoning tools integrated with Isabelle/HOL. We also address (...) the relation of our axiom systems to alternative proposals from the literature, including an axiom set proposed by Freyd and Scedrov for which we reveal a technical issue (when encoded in free logic where free variables range over defined and undefined objects): either all operations, e.g. morphism composition, are total or their axiom system is inconsistent. The repair for this problem is quite straightforward, however. (shrink)

This paper studies the structure of semantic theories over modular computational systems and applies the algebraic Theory of Institutions to provide a logical representation of such theories. A modular semantic theory is here defined by a cluster of semantic theories, each for a single program's module, and by a set of relations connecting models of different semantic theories. A semantic theory of a single module is provided in terms of the set of ∑‐models mapped from the category (...) Th of ∑‐theories and generating a hierarchy of structures from an abstract model to a concrete model of data. The collection of abstract models representing different modules of a program is formalised as the category of institutions INS, where theory morphisms express refinements, integrations, and compositions between couples of modules. Finally, it is required that a morphism in INS at any level occurs iff the same morphism occurs at the lower level alongside the Th hierarchy. (shrink)

Recent semantic approaches to scientific structuralism, aiming to make precise the concept of shared structure between models, formally frame a model as a type of set-structure. This framework is then used to provide a semantic account of (a) the structure of a scientific theory, (b) the applicability of a mathematical theory to a physical theory, and (c) the structural realist’s appeal to the structural continuity between successive physical theories. In this paper, I challenge the idea that, to (...) be so used, the concept of a model and so the concept of shared structure between models must be formally framed within a single unified framework, set-theoretic or other. I first investigate the Bourbaki-inspired assumption that structures are types of set-structured systems and next consider the extent to which this problematic assumption underpins both Suppes’ and recent semantic views of the structure of a scientific theory. I then use this investigation to show that, when it comes to using the concept of shared structure, there is no need to agree with French that “without a formal framework for explicating this concept of ‘structure-similarity’ it remains vague, just as Giere’s concept of similarity between models does ...” (French, 2000, Synthese, 125, pp. 103–120, p. 114). Neither concept is vague; either can be made precise by appealing to the concept of a morphism, but it is the context (and not any set-theoretic type) that determines the appropriate kind of morphism. I make use of French’s (1999, From physics to philosophy (pp. 187–207). Cambridge: Cambridge University Press) own example from the development of quantum theory to show that, for both Weyl and Wigner’s programmes, it was the context of considering the ‘relevant symmetries’ that determined that the appropriate kind of morphism was the one that preserved the shared Lie-group structure of both the theoretical and phenomenological models. (shrink)

We establish a criterion for deciding whether a class of structures is the class of models of a geometric theory inside Grothendieck toposes; then we specialize this result to obtain a characterization of the infinitary first-order theories which are geometric in terms of their models in Grothendieck toposes, solving a problem posed by Ieke Moerdijk in 1989.

The electron double-slit interference is re-examined from the point of view of temporal topos. Temporal topos (or t-topos) is an abstract algebraic (categorical) method using the theory of sheaves. A brief introduction to t-topos is given. When the structural foundation for describing particles is based on t-topos, the particle-wave duality of electron is a natural consequence. A presheaf associated with the electron represents both particle-like and wave-like properties depending upon whether an object in the site (t-site) is specified (particle-like) (...) or not (wave-like). It is shown that the localization of the electron at one of the slits is equivalent to choosing a particular object in the t-site and that the electron behaves as a wave when it passes through a double-slit because there are more than one object in the t-site. Also, the single-slit diffraction is interpreted as a result of the possibility of many different ways of factoring a morphism between two objects. (shrink)

Throughout the twentieth century, the automation of formal logics in computers has created unprecedented potential for practical applications of logic—most prominently the mechanical verification of mathematics and software. But the high cost of these applications makes them infeasible but for a few flagship projects, and even those are negligible compared to the ever-rising needs for verification. One of the biggest challenges in the future of logic will be to enable applications at much larger scales and simultaneously at much lower costs. (...) This will require a far more efficient allocation of resources. Wherever possible, theoretical and practical results must be formulated generically so that they can be instantiated to arbitrary logics; this will allow reusing results in the face of today’s multitude of application-oriented and therefore diverging logical systems. Moreover, the software engineering problems concerning automation support must be decoupled from the theoretical problems of designing logics and calculi; this will allow researchers outside or at the fringe of logic to contribute scalable logic-independent tools. Anticipating these needs, the author has developed the Mmt framework. It offers a modern approach towards defining, analyzing, implementing, and applying logics that focuses on modular design and logic-independent results. This paper summarizes the ideas behind and the results about Mmt. It focuses on showing how Mmt. provides a theoretical and practical framework for the future of logic. (shrink)

We describe a method for presenting (a topos closely related to) either of Freyd’s topos-theoretic models for the independence of the axiom of choice as the classifying topos for a geometric theory. As an application, we show that no such topos can admit a geometric morphism from a two-valued topos satisfying countable dependent choice.

This paper is a critical response to Andreas Bartels’ sophisticated defense of a structural account of scientific representation. We show that, contrary to Bartels’ claim, homomorphism fails to account for the phenomenon of misrepresentation. Bartels claims that homomorphism is adequate in two respects. First, it is conceptually adequate, in the sense that it shows how representation differs from misrepresentation and non-representation. Second, if properly weakened, homomorphism is formally adequate to accommodate misrepresentation. We question both claims. First, we show that homomorphism (...) is not the right condition to distinguish representation from misrepresentation and non-representation: a “representational mechanism” actually does all the work, and it is independent of homomorphism – as of any structural condition. Second, we test the claim of formal adequacy against three typical kinds of inaccurate representation in science which, by reference to a discussion of the notorious billiard ball model, we define as abstraction, pretence, and simulation. We first point out that Bartels equivocates between homomorphism and the stronger condition of epimorphism, and that the weakened form of homomorphism that Bartels puts forward is not a morphism at all. After providing a formal setting for abstraction, pretence and simulation, we show that for each morphism there is at least one form of inaccurate representation which is not accommodated. We conclude that Bartels’ theory – while logically laying down the weakest structural requirements – is nonetheless formally inadequate in its own terms. This should shed serious doubts on the plausibility of any structural account of representation more generally. (shrink)

We consider morphisms between binary relations that are used in the theory of cardinal characteristics. In [8] we have shown that there are pairs of relations with no Borel morphism connecting them. The reason was a strong impact of the first of the two functions that constitute a morphism, the so-called function on the questions. In this work we investigate whether the second half, the function on the answers' side, has a similarly strong impact. The main question (...) is: Does the nonexistence of a Borel morphism imply the non-existence of a morphism that is only Borel on the answers' side? We give sufficient conditions for an affirmative answer. The results are applied to the unsplitting relations where it has been open whether there is a morphism that is Borel on the answers' side. (shrink)

This paper is just a comment to the impressive work by A. C. Ehresmann and J.-P. Vanbremeersch on the theory of Memory Evolutive Systems (MES). MES are truly higher order systems. Hyperstructures represent a new concept which I introduced in order to capture the essence of what a higher order structure is—encompassing hierarchies and emergence. Hyperstructures are motivated by cobordism theory in topology and higher category theory. The morphism concept is replaced by the concept of a (...) bond. In the paper I briefly introduce hyperstructures motivated geometrically and suggest further developments of the MESs along these lines, which could widen up new areas of applications. (shrink)

Although the category of instinct is inscribed in almost every discourse in which the behavior of animals is analyzed, a closer look at the phenomenon that creates its specificity reveals its elusiveness. One gets the impression that something like instinctive behavior is simply not significantly different from other non-instinctive behavior. First, instinct as an innate, pre-programmed behavior disappears when its characteristics are compared with those of other behaviors acquired through learning. Secondly, the genesis of the instinct, which should be sought (...) in epigenetics and the mechanisms of individual development, shows that it does not differ fundamentally from other behaviors not inherited by genetics. Third, the best explanation for the origin of instinctive behavior is the ecological theory originally formulated by Eleanor Gibson and James J. Gibson for perception, as it allows us to understand the role of environmental factors in the formation of instinctive action patterns. (shrink)

This paper studies categories and functors in the context of reverse and computable mathematics. In ordinary reverse mathematics, we only focuses on categories whose objects and morphisms can be represented by natural numbers. We first consider morphism sets of categories and prove several associated theorems equivalent to $$\mathrm ACA_{0}$$ over the base system $$\mathrm RCA_{0}$$. The Yoneda Lemma is a basic result in category theory and homological algebra. We then develop an effective version of the Yoneda Lemma in (...) $$\mathrm RCA_{0}$$ ; as an application, we formalize an effective version of the Yoneda Embedding in $$\mathrm RCA_{0}$$. Products and coproducts are basic notions for defining special categories like semi-additive categories and additive categories. We study properties of products and coproducts of a sequence of objects of categories and provide effective characterizations of semi-additive categories and additive categories in terms of products and coproducts. Finally, we further consider the strength of theorems of category theory that are studied in this paper by methods of higher-order reverse mathematics. (shrink)

This paper presents a new reconstruction of Wittgenstein’s famous (and controversial) rule-following arguments. Two are the novel features offered by our reconstruction. In the first place, we propose a shift of the central focus of the discussion, from the general semantics and the philosophy of mind to the philosophy of mathematics and the rejection of the notion of a function. The second new feature is positive: we argue that Wittgenstein offers us a new alternative notion of a rule (to replace (...) the rejected functions), a notion reminiscent of Category Theory’s notion of a morphism. (shrink)

In 1922, Thoralf Skolem introduced the term of «relativity» as to infinity от set theory. Не demonstrated Ьу Zermelo 's axiomatics of set theory (incl. the axiom of choice) that there exists unintended interpretations of anу infinite set. Тhus, the notion of set was also «relative». We сan apply his argurnentation to Gödel's incompleteness theorems (1931) as well as to his completeness theorem (1930). Then, both the incompleteness of Реапо arithmetic and the completeness of first-order logic tum out (...) to bе also «relative» in Skolem 's sense. Skolem 's «relativity» argumentation of that kind сan bе applied to а vету wide range of problems and one сan spoke of the relativity of discreteness and continuity or, of finiiteness and infinity, or, of Cantor 's kinds of infinities, etc. The relativity of Skolemian type helps us for generaIizing Einstein 's principle of relativity from the invariance of the physical laws toward diffeomorphisms to their invariance toward anу morphisms (including and especiaIly the discrete ones). Such а kind of generalization from diffeomorphisms (then, the notion of velocity always makes sense) to anу kind of morphism (when 'velocity' mау оr maу not make sense) is an extension of the general Skolemian type оГ relativity between discreteness and continuity от between finiteness and infinity. Particularly, the Lorentz invariance is not valid in general because the notion of velocity is limited to diffeomorphisms. [п the case of entanglement, the physical interaction is discrete0. 'Velocity" and consequently, the 'Lorentz invariance'"do not make sense. Тhat is the simplest explanation ofthe argurnent EPR, which tums into а paradox оnJу if the universal validity of 'velocity' and 'Lогелtz invariance' is implicitly accepted. (shrink)

In this paper, I will be arguing that the basic infrastructure of an ineffable formal identity between name and object which is presented in the Tractatus is still very much involved in Wittgenstein's early development of the concept of grammar. First, it will be necessary to clearly describe how the identity between name and object is initially formulated in the Tractatus. Hence, in section 1, I will show how the 'picture theory' is ontologically grounded on the identity of linguistics' (...) and worldly atomic structural elements. I will discuss the ‘picture theory’ only briefly, since my main interest is to illuminate how that infrastructure remains a core aspect of Wittgenstein's “middle period” thinking: that is, in what way the identity of name and object is contained and presupposed within his concept of grammar and how it is still used as a condition for our symbolism to make sense. Another way to describe this paper's aim, this time from its end backwards, would be to say that it is to reveal that grammatical systems of rules are nothing other than the implementations of that special kind of identity, for the latter is always and already manifest within our symbolism. (shrink)

It is shown that a locally finite polyadic algebra on an infinite set V of variables is a Boolean-algebra object, endowed with some internal supremum morphism, in the category of locally finite transformation sets on V. Then, this new categorical definition of polyadic algebras is used to simplify the theory of these algebras. Two examples are given: the construction of dilatations and the definition of terms and constants.

Working in the fragment of Martin-Löfs extensional type theory [12] which has products (but not sums) of dependent types, we consider two additional assumptions: firstly, that there are (strong) equality types; and secondly, that there is a type which is universal in the sense that terms of that type name all types, up to isomorphism. For such a type theory, we give a version of Russell's paradox showing that each type possesses a closed term and (hence) that all (...) terms of each type are provably equal. We consider the kind of category theoretic structure which corresponds to this kind of type theory and obtain a categorical version of the paradox. A special case of this result is the degeneracy of a locally cartesian closed category with a morphism which is generic in the sense that every other morphism in the category can be obtained from it via pullback. (shrink)

The study of pairs of modules (over a Dedekind domain) arises from two different perspectives, as a starting step in the analysis of tuples of submodules of a given module, or also as a particular case in the analysis of Abelian structures made by two modules and a morphism between them. We discuss how these two perspectives converge to pairs of modules, and we follow the latter one to obtain an alternative approach to the classification of pairs of torsionfree (...) objects. Then we restrict our attention to pairs of free modules. Our main results are that the theory of pairs of free Abelian groups is co-recursively enumerable, and that a few remarkable extensions of this theory are decidable. (shrink)

We introduce the equational notion of a monadic bounded algebra (MBA), intended to capture algebraic properties of bounded quantification. The variety of all MBA's is shown to be generated by certain algebras of two-valued propositional functions that correspond to models of monadic free logic with an existence predicate. Every MBA is a subdirect product of such functional algebras, a fact that can be seen as an algebraic counterpart to semantic completeness for monadic free logic. The analysis involves the representation of (...) MBA's as powerset algebras of certain directed graphs with a set of "marked" points. It is shown that there are only countably many varieties of MBA's, all are generated by their finite members, and all have finite equational axiomatisations classifying them into fourteen kinds of variety. The universal theory of each variety is decidable. Finitely generated MBA's are shown to be finite, with the free algebra on r generators having exactly $2^{3.2^{r.} } ^{2^{{\rm{2}}^{{\rm{r - 1}}} } } $ elements. An explicit procedure is given for constructing this freely generated algebra as the powerset algebra of a certain marked graph determined by the number r. (shrink)

The technique of covers is now well established in semigroup theory. The idea is, given a semigroup S, to find a semigroup having a better understood structure than that of S, and an onto morphism of a specific kind from to S. With the right conditions on , the behaviour of S is closely linked to that of . If S is finite one aims to choose a finite . The celebrated results for inverse semigroups of McAlister in (...) the 1970s form the flagship of this theory.Weakly left quasi-ample semigroups form a quasivariety (of algebras of type(2, 1)), properly containing the classes of groups, and of inverse, left ample, and weakly left ample semigroups. We show how the existence of finite proper covers for semigroups in this quasivariety is a consequence of Ashs powerful theorem for pointlike sets. Our approach is to obtain a cover of a weakly left quasi-ample semigroup S as a subalgebra of S × G, where G is a group. It follows immediately from the fact that weakly left quasi-ample semigroups form a quasivariety, that is weakly left quasi-ample. We can then specialise our covering results to the quasivarieties of weakly left ample, and left ample semigroups. The latter have natural representations as (2, 1)-subalgebras of partial (one-one) transformations, where the unary operation takes a transformation to the identity map in the domain of . In the final part of this paper we consider representations of weakly left quasi-ample semigroups. (shrink)

This study consists of two parts. The first is an examination of the hermeneutical presuppositions underlying the theory of models that Moshe Idel has applied to the study of Jewish mysticism. Idel has opted for a typological approach based on multiple explanatory models, a methodology that purportedly proffers a polychromatic as opposed to a monochromatic orientation associated with Scholem and the so-called school based on his teachings. The three major models delineated by Idel are the theosophical-theurgical, the ecstatic, and (...) the magical or talismanic. Idel’s hermeneutic rests on the assumption that the phenomenon of Jewish mysticism (as the phenomenon of religion more generally) cannot be essentialized, and therefore no one methodological approach should be privileged as the exclusive means to ascertain it. In the second part of this study, I raise the possibility that affirming set patterns of thought and a unified system of symbols that link together kabbalists from different historical periods might not inevitably implicate the scholar in a methodological reductionism. Moving beyond a binary logic, which is still operative in the postmodern dichotomy of truth and dissimulation, I surmise that the polysemic nature of the text that may be elicited from kabbalistic sources is not dependent on the rejection of laying claim to an inherent and original intent that is recoverable through proper philological attunement. Multivocality and essentialism are not mutually exclusive. Kabbalah, I submit, is a cultural-literary phenomenon that illustrates an open system in which each moment is a mix of newness and repetition, each event a renewed singularity. The hermeneutical praxis appropriate to this system displays a temporality linked to the conception of time in its most rudimentary form as an instant of diremptive reiteration, the repetition of the same as different in the renewal of the different as same. The tendency to generalize, therefore, should not be misconstrued as viewing the variegated history of Jewish mystical doctrines and practices monolithically. (shrink)

To properly understand behavior, we must obtain both ultimate and proximate explanations. Put briefly, ultimate explanations are concerned with why a behavior exists, and proximate explanations are concerned with how it works. These two types of explanation are complementary and the distinction is critical to evolutionary explanation. We are concerned that they have become conflated in some areas of the evolutionary literature on human behavior. This article brings attention to these issues. We focus on three specific areas: the evolution of (...) cooperation, transmitted culture, and epigenetics. We do this to avoid confusion and wasted effort—dangers that are particularly acute in interdisciplinary research. Throughout this article, we suggest ways in which misunderstanding may be avoided in the future. (shrink)

Since it was first published, Quantum Field Theory in a Nutshell has quickly established itself as the most accessible and comprehensive introduction to this profound and deeply fascinating area of theoretical physics. Now in this fully revised and expanded edition, A. Zee covers the latest advances while providing a solid conceptual foundation for students to build on, making this the most up-to-date and modern textbook on quantum field theory available. -/- This expanded edition features several additional chapters, as (...) well as an entirely new section describing recent developments in quantum field theory such as gravitational waves, the helicity spinor formalism, on-shell gluon scattering, recursion relations for amplitudes with complex momenta, and the hidden connection between Yang-Mills theory and Einstein gravity. Zee also provides added exercises, explanations, and examples, as well as detailed appendices, solutions to selected exercises, and suggestions for further reading. (shrink)

Major shifts in the field of model theory in the twentieth century have seen the development of new tools, methods, and motivations for mathematicians and philosophers. In this book, John T. Baldwin places the revolution in its historical context from the ancient Greeks to the last century, argues for local rather than global foundations for mathematics, and provides philosophical viewpoints on the importance of modern model theory for both understanding and undertaking mathematical practice. The volume also addresses the (...) impact of model theory on contemporary algebraic geometry, number theory, combinatorics, and differential equations. This comprehensive and detailed book will interest logicians and mathematicians as well as those working on the history and philosophy of mathematics. (shrink)

This dissertation aims to provide a comprehensive account of set theory with urelements. In Chapter 1, I present mathematical and philosophical motivations for studying urelement set theory and lay out the necessary technical preliminaries. Chapter 2 is devoted to the axiomatization of urelement set theory, where I introduce a hierarchy of axioms and discuss how ZFC with urelements should be axiomatized. The breakdown of this hierarchy of axioms in the absence of the Axiom of Choice is also (...) explored. In Chapter 3, I investigate forcing with urelements and develop a new approach that addresses a drawback of the existing machinery. I demonstrate that forcing can preserve, destroy, and recover the axioms isolated in Chapter 2 and discuss how Boolean ultrapowers can be applied in urelement set theory. Chapter 4 delves into class theory with urelements. I first discuss the issue of axiomatizing urelement class theory and then explore the second-order reflection principle with urelements. In particular, assuming large cardinals, I construct a model of second-order reflection where the principle of limitation of size fails. (shrink)

The Handicap Principle represents a central theory in the biological understanding of signaling. This paper presents a number of alternative theories to the Handicap Principle and argues that some of these theories may provide a better explanation for the evolution and stability of honest communication.

In the past decade the central principles of western feminist theory have been dramatically challenged. many feminists have endorsed post-structuralism's rejection of essentialist theoretical categories, and have added a powerful gender dimension to contemporary critiques of modernity. Earlier 'women' have been radically undermined, and newer concerns with 'difference', 'identity', and 'power' have emerged. Destabilizing Theory explores these developments in a set of specially commissioned essays by feminist theorists. Does this change amount to a real shift within feminist (...) class='Hi'>theory, or will feminism's links with an emancipatory modernism reinstate an older political agenda? Can we transcend the common counterposition of equality and difference, or is feminism condemned to argue within the terms of this binary opposition? (shrink)

Traditionally, epistemologists have distinguished between epistemic and pragmatic goals. In so doing, they presume that much of game theory is irrelevant to epistemic enterprises. I will show that this is a mistake. Even if we restrict attention to purely epistemic motivations, members of epistemic groups will face a multitude of strategic choices. I illustrate several contexts where individuals who are concerned solely with the discovery of truth will nonetheless face difficult game theoretic problems. Examples of purely epistemic coordination problems (...) and social dilemmas will be presented. These show that there is a far deeper connection between economics and epistemology than previous appreciated. (shrink)