This book explains, in lay terms, the surprisingly simple system of mathematical logic used in digital computer circuitry. Anecdotal in its style and often funny, it follows the development of this logic system from its origins in Victorian England to its rediscovery in this century as the foundation of all modern computing machinery. ONES AND ZEROS will be enjoyed by anyone who has a general interest in science and technology.

The invention of symbolic algebra in the sixteenth and seventeenth centuries fundamentally changed the way we do mathematics. If we want to understand this change and appreciate its importance, we must analyze it on two levels. One concerns the compositional function of algebraic symbols as tools for representing complexity; the other concerns the referential function of algebraic symbols, which enables their use as tools for describing objects (such as polynomials), properties (such as irreducibility), relations (such as divisibility), and (...) operations (such as factorization). The reconstruction of both the compositional function and the referential function of algebraic symbols requires the use of different analytic tools and the taking of different temporal perspectives. In this chapter, we offer both: a reconstruction of the compositional function of algebraic symbols, and a reconstruction of their referential function. (shrink)

By a sentential logic we understand a pair, where S is a sentential language, i.e. an absolutely free algebra freely generated by an infinite set p, q, r,... of sentential variables and endowed with countably many finitary connectives §1, §2,... and C is a consequence operation on S, the underlying set of S, satisfying the condition of structurality: eC ⊆ C, for every endomorphism e of S and for every X ⊆ S. If no confusion is likely we shall identify (...) a logic with its consequence operation C. A logic C is standard if C = [ {C : Y ⊆ X & Y is finite}, for all X ⊆ S. (shrink)

We present an algebraic account of the Tongan kinship terminology (TKT) that provides an insightful journey into the fabric of Tongan culture. We begin with the ethnographic account of a social event. The account provides us with the activities of that day and the centrality of kin relations in the event, but it does not inform us of the conceptual system that the participants bring with them. Rather, it is a slice in time of an ongoing dynamic process that (...) links behavior with a conceptual system of kin relations and vice versa. To understand this interplay, we need an account of the underlying conceptual system that is being activated during the event. Thus, we introduce a formal, algebraically based account of TKT. This account brings to the fore the underlying logic of TKT and allows us to distinguish between features of the kinship system that arise from the logic of TKT as a generative structure and features that must have arisen through cultural intervention. (shrink)

In the present paper we study how subsystems of a normative system can be combined, and the role of such combinations for the understanding of hypothetical legal consequences. A combination of two subsystems is often accomplished by a normative correlation or an intermediate concept. To obtain a detailed analysis of such phenomena we use an algebraic framework. Normative systems are represented as algebraic structures over sets of conditions. This representation makes it possible to study normative systems using (...) an extension of the theory of Boolean algebras, called the theory of Boolean quasi‐orderings. (shrink)

El problema fundamental para que la Teoría de la relatividad pueda ser acorde con la filosofía de Kant es el de la utilización de una geometría no euclídea. Que sus principios sean interpretados como juicios sintéticos a priori es, a nuestro entender, un problema secundario. Si queremos que los principios de una ciencia de la naturaleza sean universales y necesarios sin recurrir a dogmatismos, no queda otra posibilidad que entenderlos trascendentalmente. Como se observa en el principio de relatividad, Einstein también (...) pensaba que las leyes físicas son universales y necesarias. Sin embargo, su perspectiva fue más racionalista que crítica. Sea como fuere, si los principios de la Teoría de la relatividad son a priori y enlazan sintéticamente la experiencia posible, esta teoría nos servirá para entender que las geometrías no euclídeas no son ajenas a los planteamientos de Kant, si bien requieren dar un rodeo epistemológico. La doctrina que nos permite fundamentar las geometrías no euclidianas desde la filosofía de Kant la llamamos “Álgebra de la experiencia”.The main problem for the Theory of the relativity to be in agreement with the Kant´s philosophy is the use of a non-Euclidean geometry. The fact that their principles could be interpreted a priori as synthetic trials is, in our opinion, a secondary problem. If we want that the principles of a science of the nature would be universal and necessary ones without appealing to dogmatisms, then, he have to understand them transcendentally. As in the principle of relativity it is observed, Einstein also thought that the physical laws are universal and necessary ones. However, their perspective was more rationalistic than critic. In any case, if the Theory of Relativity principles are a priori and they synthetically connect the possible experience, then, this theory will allow us to understand that the non-Euclidean geometries are not far away from the Kant´s positions, although they require to made a epistemological turn around. The principle that allows us to base the geometries non Euclidian from the Kant´s philosophy is called “Algebra of the experience”. (shrink)

This book presents the foundations of a general theory of algebras. Often called “universal algebra”, this theory provides a common framework for all algebraic systems, including groups, rings, modules, fields, and lattices. Each chapter is replete with useful illustrations and exercises that solidify the reader's understanding. The book begins by developing the main concepts and working tools of algebras and lattices, and continues with examples of classical algebraic systems like groups, semigroups, monoids, and categories. The essence of (...) the book lies in Chapter 4, which provides not only basic concepts and results of general algebra, but also the perspectives and intuitions shared by practitioners of the field. The book finishes with a study of possible uniqueness of factorizations of an algebra into a direct product of directly indecomposable algebras. There is enough material in this text for a two semester course sequence, but a one semester course could also focus primarily on Chapter 4, with additional topics selected from throughout the text. (shrink)

The desire to understand the mathematics of living systems is increasing. The widely held presupposition that the mathematics developed for modeling of physical systems as continuous functions can be extended to the discrete chemical reactions of genetic systems is viewed with skepticism. The skepticism is grounded in the issue of scientific invariance and the role of the International System of Units in representing the realities of the apodictic sciences. Various formal logics contribute to the theories of biochemistry and molecular biology (...) and genetics. Various paths of extension are invoked in these formal logics in order to express the information of biological apodicticism. Symbolizing the appropriate notations for invariant relations and for biological extensions of relations is fundamental to the exact generating functions of discrete algebraic biology. Aspects of philosophical perspectives of the relation scientific number systems are contrasted. The deep distinction between physical motion and biological motion is expressed in terms the roles of Aristotelian causes. The interior motion within perplex numbers is contrasted with the exterior motion of physical systems. The need for a new mathematics for biology is suggested. (shrink)

'The Greate Invention of Algebra' casts new light on the work of Thomas Harriot, an innovative thinker and practitioner in several branches of the mathematical sciences, including navigation, astronomy, optics, geometry, and algebra. Although on his death Harriot left behind over four thousand manuscript sheets, much of his work remains unpublished. This book focuses on one hundred and forty of Harriot's manuscript pages, those concerned with the structure and solution of equations. The original material has been carefully ordered, translated, and (...) annotated to provide the first complete edition of his work on this subject, and an extended introduction provides the reader with a lucid background to the work and explains its contents. Illustrations from the manuscripts provide fascinating reference material. The appendix discusses correlations between Harriot's manuscripts and the texts of his contemporaries Viète, Warner, and Torporley. The clear and concise exposition makes this an excellent reference volume for historians of mathematics and those interested in the history of science. This is an important new resource for understanding the development of algebra in seventeenth-century England. (shrink)

Based on existence equations, quasivarieties of heterogeneous partial algebras have the same algebraic description as those of total algebras. Because of the restriction of the valuations to the free variables of a formula — the usual reference to the needed variables e.g. for identities (in order to get useful and manageable results) is essentially replaced here by the use of the logical Craig projections — already varieties of heterogeneous partial algebras behave to some extent rather like quasivarieties than having (...) the properties known from varieties of total homogeneous algebras. It is one of the main aims of this note to make this more explicit. On the other hand we want to list several results known for quasivarieties of heterogeneous partial algebras — and adopt them to the extended signature — after having recalled the language and the main concepts necessary for the understanding of the results. (shrink)

Based on existence equations, quasivarieties of heterogeneous partial algebras have the same algebraic description as those of total algebras. Because of the restriction of the valuations to the free variables of a formula — the usual reference to the needed variables e.g. for identities (in order to get useful and manageable results) is essentially replaced here by the use of the “logical Craig projections” — already varieties of heterogeneous partial algebras behave to some extent rather like quasivarieties than having (...) the properties known from varieties of total homogeneous algebras. It is one of the main aims of this note to make this more explicit. On the other hand we want to list several results known for quasivarieties of heterogeneous partial algebras — and adopt them to the extended signature — after having recalled the language and the main concepts necessary for the understanding of the results. (shrink)

The semantics of directional prepositions is investigated from the perspective of aspect. What distinguishes telic PPs (like to the house) from atelic PPs (like towards the house), taken as denoting sets of paths, is their algebraic structure: atelic PPs are cumulative, closed under the operation of concatenation, telic PPs are not. Not only does this allow for a natural and compositional account of how PPs contribute to the aspect of a sentence, but it also guides our understanding of (...) the lexical semantics of prepositions in important ways. Semantically, prepositions turn out to be quite similar to nouns and verbs. Nominal distinctions (like singular and plural, mass and count) and verbal classes (like semelfactives and degree achievements) have their prepositional counterparts. (shrink)

“If the tower is any taller than 320 ms, it may collapse,” Eiffel thinks out loud. Although understanding this counterfactual poses no trouble, the most successful interventionist semantics struggle to model it because the antecedent can come about in infinitely many ways. My aim is to provide a semantics that will make modeling such counterfactuals easy for philosophers, computer scientists, and cognitive scientists who work on causation and causal reasoning. I first propose three desiderata that will guide my theory: (...) it should be general, yet conservative, yet useful. Next, I develop a formalization of events in the form of an algebra. I identify an event with all the ways in which it can be brought about and provide rules for determining the referent of an arbitrary event description. I apply this algebra to counterfactuals expressed using underdeterministic causal models-models that encode non-probabilistic causal indeterminacies. Specifically, I develop semaphore interventions, which represent how the target system may be modified from without in a coordinated fashion. This, in turn, allows me to bring about any event within a single model. Finally, I explain the advantages of this semantics over other interventionist competitors. (shrink)

Daniel Sutherland - Kant on Arithmetic, Algebra, and the Theory of Proportions - Journal of the History of Philosophy 44:4 Journal of the History of Philosophy 44.4 533-558 Muse Search Journals This Journal Contents Kant on Arithmetic, Algebra, and the Theory of Proportions Daniel Sutherland Kant's philosophy of mathematics has both enthralled and exercised philosophers since the appearance of the Critique of Pure Reason. Neither the Critique nor any other work provides a sustained and focused account of his mature views (...) on mathematical cognition, forcing readers to glean what they can from disparate contexts. Despite these hurdles, Kant's views have been of great interest to philosophers of mathematics. They have also been of interest to philosophers wishing to understand Kant's philosophy more generally, since Kant maintains that mathematical judgments are synthetic a priori and that the type of synthesis underlying mathematics is the same as that underlying the perception of objects . Our understanding of Kant's views has been greatly improved during the last four decades by work on Kant's philosophy of mathematics. Despite these advances, however, I think the recent work has largely neglected the importance of Kant's theory of magnitudes and the extent to which that theory is influenced by the Eudoxian theory of proportions presented in Euclid's Elements. As a consequence, an important feature of Kant's.. (shrink)

The article offers an analysis of Simone Weil's philosophy of mathematics. Weil's reflection starts from a critique of Bourbaki's programme, led by her brother André: the "mechanical attention" Bourbaki considered an advantage of their treatment of mathematics was for her responsible for the incomprehensibility of modern algebra, and even a cause of alien-ation and social oppression. On the contrary, she developed her pivotal concept of 'atten-tion' with the aim of approaching mathematical problems in order to make "progress in another more (...) mysterious dimension". In the Pythagorean 'crisis of incommensurables', Weil saw the possibility of defining the relationships between things in terms that are not exclusively numerical. This implies drawing an analogy between mathematical relation-ships and God's relationship with mankind (logos), the basis of a 'supernatural' reformu-lation of the entire scientific understanding of the world. The consequence is a critique of machinism and the possibility to contrast algorithmic reason with a "supernatural reason". (shrink)

Deontic logic is devoted to the study of logical properties of normative predicates such as permission, obligation and prohibition. Since it is usual to apply these predicates to actions, many deontic logicians have proposed formalisms where actions and action combinators are present. Some standard action combinators are action conjunction, choice between actions and not doing a given action. These combinators resemble boolean operators, and therefore the theory of boolean algebra offers a well-known athematical framework to study the properties of the (...) classic deontic operators when applied to actions. In his seminal work, Segerberg uses constructions coming from boolean algebras to formalize the usual deontic notions. Segerberg’s work provided the initial step to understand logical properties of deontic operators when they are applied to actions. In the last years, other authors have proposed related logics. In this chapter we introduce Segerberg’s work, study related formalisms and investigate further challenges in this area. (shrink)

This article reconstructs the theory of the Soviet-American psychologist Vladimir Lefebvre as part of the neocybernetic movement. In particular, I propose to explore such elements of his research of the 1970s—1990s as systemic vision; reflexive analysis; a search for holistic configuration and Janus cosmology; and the realization of neocybernetics. An interest in the reflexive structures of cognition and action led Lefebvre to an understanding of the limited nature of the world’s scientific picture. The conflicting objects he studied proved too (...) complex for behavioral modeling and forecasting. Lefebvre therefore expanded the boundaries of the application of his formal theory of intelligence, relying on functionalist positions, which led him to the science-fiction narrative of “The Big Correction.” I contend that Lefebvre’s works not only propose an original Soviet version of the neocybernetics the 1970s but develop their own model of inhumanism long before it appeared on the scene of modern philosophy. Lefebvre’s formal theory, reinforced by thermodynamics and elements of Russian cosmism, builds cognitive behavior into the teleology of the Big Correction, which involves the idea of overcoming the heat death of the universe. The activity of cosmic intelligence, which Lefebvre equates with magnetic-plasma structures in the finished areas of the universe, is aimed at cooperation and prolonging the existence of life and the intellect. (shrink)

We approach the relationship between classical and quantum theories in a new way, which allows both to be expressed in the same mathematical language, in terms of a matrix algebra in a phase space. This makes clear not only the similarities of the two theories, but also certain essential differences, and lays a foundation for understanding their relationship. We use the Wigner-Moyal transformation as a change of representation in phase space, and we avoid the problem of “negative probabilities” by (...) regarding the solutions of our equations as constants of the motion, rather than as statistical weight factors. We show a close relationship of our work to that of Prigogine and his group. We bring in a new nonnegative probability function, and we propose extensions of the theory to cover thermodynamic processes involving entropy changes, as well as the usual reversible processes. (shrink)

This paper surveys the issue of relativistic causality within the framework of algebraic quantum field theory . In doing so, we distinguish various notions of causality formulated in the literature and study their relationships, and thereby we offer what we hope to be a useful taxonomy. We propose that the most direct expression of relativistic causality in AQFT is captured not by the spectrum condition but rather by the axiom of local primitive causality, in that it entails a form (...) of local determinism for quantum fields which generalizes the constraint of no superluminal propagation of classical field theories to relativistic quantum field theory. We discuss the status of the axiom of micro-causality by locating its place within a large family of separability/independence/locality conditions developed for AQFT and also by relating it to so-called no-signalling theorems. And we also provide a critical survey of attempts to understand the implications for relativistic causality of the distant correlations endemic to the states in models of AQFT satisfying the standard axioms, and we provide an assessment of attempts to employ Reichenbach's common cause principle in AQFT to defuse worries that these distant correlations implicate direct causal connections between relatively spacelike events. (shrink)

In this paper the following theorem is proved regarding groups of finite Morley rank which are perfect central extensions of quasisimple algebraic groups.Theorem1.Let G be a perfect group of finite Morley rank and let C0be a definable central subgroup of G such that G/C0is a universal linear algebraic group over an algebraically closed field; that is G is a perfect central extension of finite Morley rank of a universal linear algebraic group. Then C0= 1.Contrary to an impression (...) which exists in some circles, the center of the universal extension of a simple algebraic group, as an abstract group, is not finite in general. Thus the finite Morley rank assumption cannot be omitted.Corollary1.Let G be a perfect group of finite Morley rank such that G/Z(G) is a quasisimple algebraic group. Then G is an algebraic group. In particular, Z(G) is finite([4],Section27.5).An understanding of central extensions of quasisimple linear algebraic groups which are groups of finite Morley rank is necessary for the classification oftamesimpleK*-groupsof finite Morley rank, which constitutes an approach to the Cherlin-Zil’ber conjecture. For this reason the theorem above and its corollary were proven in [1] (Theorems 4.1 and 4.2) under the assumption oftameness, which simplifies the argument considerably. The result of the present paper shows that this assumption can be dropped. The main line of argument is parallel to that in [1]; the absence of the tameness assumption will be countered by a model-theoretic result and results fromK-theory. The model-theoretic result places limitations on definability in stable fields, and may possibly be relevant to eliminating certain other uses of tameness. (shrink)

Descartes’s description of his method for discovering truth provides a helpful tool for interpreting his writings. In this article I offer a sample of how to interpret Descartes by understanding his algebraic method. My test case is the Cartesian teaching on divine freedom, which is well known to be inconsistent and often considered unfounded. I reconstruct the equations that led to these doctrines, arguing that Descartes held that the divine act of creation was both necessary and arbitrary because (...) of the equations that resulted when he applied his method to the natural world. (shrink)

In this essay I will defend three points, the first being that Descartes- unlike the aristotelian traditon- maintained that abstraction is not a operation in which the intellect builds the mathematical object resorting to sensible ob- jects. Secondly I will demonstrate that, according to cartesian philosophy, the faculty of understanding has the ability to instatiate- within the process of abstraction- mathematical symbols that represent the relation between quantities, whether magnitude or multitude.And finally I will advocate that the lack of (...) onthological commitment with sensible experience found in cartesian philosophy of mathematics allows for the creation of a mathematical language that regards the objects of geometry and arithmetics through a system of rules and notations, in other words, algebra. (shrink)

Bundles of C*-algebras can be used to represent limits of physical theories whose algebraic structure depends on the value of a parameter. The primary example is the ℏ→0 limit of the C*-algebras of physical quantities in quantum theories, represented in the framework of strict deformation quantization. In this paper, we understand such limiting procedures in terms of the extension of a bundle of C*-algebras to some limiting value of a parameter. We prove existence and uniqueness results for such extensions. (...) Moreover, we show that such extensions are functorial for the C*-product, dynamical automorphisms, and the Lie bracket (in the ℏ→0 case) on the fiber C*-algebras. (shrink)

In [3], O. C. García and W. Taylor make an in depth study of the lattice of interpretability types of varieties first introduced by W. Neumann [5]. In this lattice several varieties are identified so in order to distinguish them and understand the fine structure of the lattice, we propose the study of the interpretations between them, in particular, how many there are and what these are. We prove, among other things, that there are eight interpretations from the variety of (...) Monadic algebras into itself. (shrink)

I discuss Earman's program to achieve an objective account of the Higgs mechanism within the C∗ algebraic approach to quantum field theory. Pointing to three results obtained within this approach, I argue that if one follows Earman and understands the Higgs mechanism as a constraint, it appears to be a genuine quantum phenomenon that does not simply arise through the correspondence principle. This casts further this casts doubts on the validity of the Dirac conjecture that identifies first-class constraints and (...) gauge transformations and that provides a major motivation for both Earman's program and Struyve's demonstration that a gauge-invariant account of the Higgs mechanism can be accomplished at the classical level. (shrink)

Logics designed to deal with vague statements typically allow algebraic semantics such that propositions are interpreted by elements of residuated lattices. The structure of these algebras is in general still unknown, and in the cases that a detailed description is available, to understand its significance for logics can be difficult. So the question seems interesting under which circumstances residuated lattices arise from simpler algebras in some natural way. A possible construction is described in this paper.Namely, we consider pairs consisting (...) of a Brouwerian algebra and an equivalence relation. The latter is assumed to be in a certain sense compatible with the partial order, with the formation of differences, and with the formation of suprema of pseudoorthogonal elements; we then call it an s-equivalence relation. We consider operations which, under a suitable additional assumption, naturally arise on the quotient set. The result is that the quotient set bears the structure of a residuated lattice. Further postulates lead to dual BL-algebras. In the case that we begin with Boolean algebras instead, we arrive at dual MV-algebras. (shrink)

In this paper we give an account of the rise and development of coalgebraic thinking in mathematics and computer science as an illustration of the way mathematical frameworks may be transformed. Originating in a foundational dispute as to the correct way to characterise sets, logicians and computer scientists came to see maximizing and minimizing extremal axiomatisations as a dual pair, each necessary to represent entities of interest. In particular, many important infinitely large entities can be characterised in terms of such (...) axiomatisations. We consider reasons for the delay in arriving at the coalgebraic framework, despite many unrecognised manifestations occurring years earlier, and discuss an apparent asymmetry in the relationship between algebra and coalgebra. (shrink)

Hohfeld’s analysis on the different types of rights and duties is highly influential in analytical legal theory, and it is considered as a fundamental theory in AI&Law and normative multi-agent systems. Yet a century later, the formalization of this theory remains, in various ways, unresolved. In this paper I provide a formal analysis of how the working of a system containing Hohfeldian rights and duties can be delineated. This formalization starts from using the same tools as the classical ones by (...) Kanger and Lindahl used, but instead of focusing on the algebraic features of rights and duties, it aims at providing a comprehensive analysis of what these rights and duties actually are and how they behave and at saying something substantial on Power too—maintaining all along the Hohfeldian intentions that these rights and duties are sui generis and inherently relational. (shrink)

This study provides an innovative approach to neutrosophic algebraic structures by introducing a new structure called Neutrosophic Soft Cubic T-ideal (NSCTID), which combines T-ideal (TID) and neutrosophic Soft Cubic Sets (NSCSs) within the framework of PS-Algebra. Within the already-existing neutrosophic cubic structures, the addition of soft sets with the characteristics of TID makes this structure more desirable. The theoretical development of the proposed structure includes the application of fundamental ideas as union, intersection, the Cartesian product, and homomorphism. We also (...) introduce the notions of NSCTID-translation and NSCTID-multiplication to further enhance the structure of NSCTID. By applying the idea of translation and multiplication, we offer improved algorithm for neutrosophic cubic sets to deal with different parameters that are satisying the TID’s distinctive characteristics. Through this thorough research, we offer an elementary understanding of NSCTID and its capabilities, providing the way to new algebraic structures. (shrink)

This paper develops an algebraic formulation summarizing various forms of socioeconomic interaction in and across individuals, groups, corporations, and states. The proposed articulation accelerates the understanding that coordination among economic agents leads to the efficient allocation of resources in society. The study considers an approach whereby the State has a regulatory role which helps attain responsible consumption and production choices (RCP). This study has the potential to encourage the use of resources in a way that promotes RCP decisions (...) based on an understanding of values. The fundamental structure of socioeconomic interaction is described in work using finite algebra. It presents how governance structure comprising intra- and interactions involving the State, corporation, groups, and the individual will attain (RCP). The prevailing economic structures functioned lower than the projected level, with enormous damage to the environment. It is deduced that the final level of the economy depends crucially on the initial injection of socioeconomic actions. Furthermore, diverse socioeconomic position patterns are possible depending on the nature of activity impulses undertaken at different levels. The world needs to decouple environmental degradation from economic growth to evolve towards a sustainable and collectively inclusive economy. For that purpose, mutual trust-based coordination among all stakeholders is vital to attain sustainable objectives. (shrink)

In this paper we present a category equivalent to that of tense Nelson algebras. The objects in this new category are pairs consisting of an IKt-algebra and a Boolean IKt-congruence and the morphisms are a special kind of IKt-homomorphisms. This categorical equivalence permits understanding tense Nelson algebras in terms of the better–known IKt-algebras.

ABSTRACTThis paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on ‘algebraical geometry’ and ‘geometrical algebra’ in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but also to (...) initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the ‘abstract algebra’ and ‘abstract geometry’ of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to ‘algebraical geometry’ and ‘geometrical algebra’ of the second generation of ‘scientific’ symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s–1840s. (shrink)

This paper continues the investigation, started in Lávička and Noguera : 521–551, 2017), of infinitary propositional logics from the perspective of their algebraic completeness and filter extension properties in abstract algebraic logic. If follows from the Lindenbaum Lemma used in standard proofs of algebraic completeness that, in every finitary logic, intersection-prime theories form a basis of the closure system of all theories. In this article we consider the open problem of whether these properties can be transferred to (...) lattices of filters over arbitrary algebras of the logic. We show that in general the answer is negative, obtaining a richer hierarchy of pairwise different classes of infinitary logics that we separate with natural examples. As by-products we obtain a characterization of subdirect representation for arbitrary logics, develop a fruitful new notion of natural expansion, and contribute to the understanding of semilinear logics. (shrink)

Alchemists' illustrations indicated through symbols the processes being attempted; but with Lavoisier's Elements (1789), the place of imagination and symbolic language in chemistry was much reduced. He sought to make chemistry akin to algebra and its illustrations merely careful depictions of apparatus. Although younger contemporaries sought, and found in electrochemistry, a dynamical approach based upon forces rather than weights, they found this very difficult to picture. Nevertheless, by looking at chemical illustrations in the eighty years after Lavoisier's revolutionary book, we (...) can learn about how reactions were carried out, and interpreted, and see that there was scope for aesthetic judgement and imagination. (shrink)

Formalism in the philosophy of mathematics has taken a variety of forms and has been advocated for widely divergent reasons. In Sects. 1 and 2, I briefly introduce the major formalist doctrines of the late nineteenth and early twentieth centuries. These are what I call empirico-semantic formalism, game formalism and instrumental formalism. After describing these views, I note some basic points of similarity and difference between them. In the remainder of the paper, I turn my attention to Hilbert’s instrumental formalism. (...) My primary aim there will be to develop its formalist elements more fully. These are, in the main, its rejection of the axiom-centric focus of traditional model-construction approaches to consistency problems, its departure from the traditional understanding of the basic nature of proof and its distinctively descriptive or observational orientation with regard to the consistency problem for arithmetic. More specifically, I will highlight what I see as the salient points of connection between Hilbert’s formalist attitude and his finitist standard for the consistency proof for arithmetic. I will also note what I see as a significant tension between Hilbert’s observational approach to the consistency problem for arithmetic and his expressed hope that his solution of that problem would dispense with certain epistemological concerns regarding arithmetic once and for all. (shrink)

ABSTRACT Ruetsche claims that an abstract C*-algebra of observables will not contain all of the physically significant observables for a quantum system with infinitely many degrees of freedom. This would signal that in addition to the abstract algebra, one must use Hilbert space representations for some purposes. I argue to the contrary that there is a way to recover all of the physically significant observables by purely algebraic methods. 1Introduction 2Preliminaries 3Three Extremist Interpretations 3.1Algebraic imperialism 3.2Hilbert space conservatism 3.3Universalism (...) 4Parochial Observables 4.1Parochial observables for the imperialist 4.2Parochial observables for the universalist 5Conclusion. (shrink)

Ruetsche claims that an abstract C*-algebra of observables will not contain all of the physically significant observables for a quantum system with infinitely many degrees of freedom. This would signal that in addition to the abstract algebra, one must use Hilbert space representations for some purposes. I argue to the contrary that there is a way to recover all of the physically significant observables by purely algebraic methods. 1 Introduction2 Preliminaries3 Three Extremist Interpretations3.1 Algebraic imperialism3.2 Hilbert space conservatism3.3 (...) Universalism4 Parochial Observables4.1 Parochial observables for the imperialist4.2 Parochial observables for the universalist5 Conclusion. (shrink)

The sum of all objects of a science, the objects’ features and their mutual relations compose the reality described by that sense. The reality described by mathematics consists of objects such as sets, functions, algebraic structures, etc. Generally speaking, the use of terms reality and existence, in relation to describing various objects’ characteristics, usually implies an employment of physical and perceptible attributes. This is not the case in mathematics. Its reality and the existence of its objects, leaving aside its (...) application, are completely virtual and yet clearly organized. This organization can be recognized in the creation of axioms or in the arrival at new theorems and definitions. It results either from a formalization of an intuitive idea or from a combinatorics that has not been guided by intuition. In all four possible cases—therefore, also in the two in which there is no intuitive “lead”, we can plausibly talk about a discovery of mathematical facts and thus support the Platonist view. (shrink)

The n-th PowerSet of a Set {or Pn(S)} better describes our real world, because a system S (which may be a company, institution, association, country, society, set of objects/plants/animals/beings, set of concepts/ideas/propositions, etc.) is formed by sub-systems, which in their turn by sub-sub-systems, and so on. We prove that the SuperHyperFunction is a generalization of classical Function, SuperFunction, and HyperFunction. And the SuperHyperAlgebra, SuperHyperGraph are part of the SuperHyperStructure. Almost all structures in our real world are Neutrosophic SuperHyperStructures since they (...) have indeterminate/incomplete/uncertain/conflicting data. (shrink)

Work on representing women's voices in ethics has produced a vision of moral understanding profoundly subversive of the traditional philosophical conception of moral knowledge. 1 explicate this alternative moral “epistemology,” identify how it challenges the prevailing view, and indicate some of its resources for a liberatory feminist critique of philosophical ethics.

In his Locke Lectures Brandom proposes to extend what he calls the project of analysis to encompass various relationships between meaning and use. As the traditional project of analysis sought to clarify various logical relations between vocabularies so Brandom’s extended project seeks to clarify various pragmatically mediated semantic relations between vocabularies. The point of the exercise in both cases is to achieve what Brandom thinks of as algebraic understanding. Because the pragmatist critique of the traditional project of analysis (...) was precisely to deny that such understanding is appropriate to the case of natural language, the very idea of an analytic pragmatism is called into question by that critique. My aim is to clarify the prospects for Brandom’s project, or at least something in the vicinity of that project, through a comparison of it with what I will suggest we can think of as Kant’s analytic pragmatism as developed by Peirce. (shrink)

Este artículo presenta a Weber y Habermas como los umbrales o polos de una modernidad que tiene al progreso como horizonte teórico-práctico. El diagnóstico weberiano sobre la modernidad y su proceso de desencantamiento del mundo y la injustificada reducción de la actividad racional a una actividad utilitario-estratégica desprovista de su carácter veritativo y de su orientación valórica, Habermas la utiliza para justificar su propuesta teórico-crítica respecto a la modernidad y la "paradoja de la racionalización", distinguiendo "sistema" y "mundo vital". Aquí (...) "desencantamiento del mundo" y "consenso del proyecto de lamodernidad", se encuentranThis article presents Weber and Habermas like the thresholds or poles of a modernity that has to the progress as theoretical- practical horizon. The diagnosis weberiano on the modernity and his process of disenchantment of the world and the unjustified reduction of the rational activity to a utilitarian-strategic activity devoid of his character of truth, Habermas uses to justify his theoretical-critical offerwith regard to themodernity and the "paradox of the rationalization", distinguishing "system" and "vital world". Here "disenchantment of the world" and "consensus of the project of the modernity", they are. (shrink)