The contribution deals with some key problems of cognitive science, whose plurality transcends the boundaries of the disciplines drawn by classical epistemology. In particular, it addresses the issues of mental images, spaces of representation, and the architecture of cognitive processes in vision theory. The thesis presented is that a proper treatment of vision within psychophysics entails an analysis of a series of interconnected spaces, objects and methodologies, from psychophysics to the many virtual realities of representation.

Visual space can be distinguished from physical space. The first is found in visual experience, while the second is defined independently of perception. Theorists have wondered about the relation between the two. Some investigators have concluded that visual space is non-Euclidean, and that it does not have a single metric structure. Here it is argued that visual space exhibits contraction in all three dimensions with increasing distance from the observer, that experienced features of this contraction are (...) not the same as would be the experience of a perspective projection onto a fronto-parallel plane, and that such contraction is consistent with size constancy. These properties of visual space are different from those that would be predicted if spatial perception resulted from the successful solution of the inverse problem. They are consistent with the notion that optical constraints have been internalized. More generally, they are also consistent with the notion that visual spatial structures bear a resemblance relation to physical spatial structures. This notion supports a type of representational relation that is distinct from mere causal correspondence. The reticence of some philosophers and psychologists to discuss the structure of phenomenal space is diagnosed in terms of the simple materialism and the functionalism of the 1970s and 1980s. (shrink)

The sets of contexts and properties of a concept are embedded in the complex Hilbert space of quantum mechanics. States are unit vectors or density operators, and contexts and properties are orthogonal projections. The way calculations are done in Hilbert space makes it possible to model how context influences the state of a concept. Moreover, a solution to the combination of concepts is proposed. Using the tensor product, a procedure for describing combined concepts is elaborated, providing a natural (...) solution to the pet fish problem. This procedure allows the modeling of an arbitrary number of combined concepts. By way of example, a model for a simple sentence containing a subject, a predicate and an object, is presented. (shrink)

Some regularities enjoy only an attenuated existence in a body of training data. These are regularities whose statistical visibility depends on some systematic recoding of the data. The space of possible recodings is, however, infinitely large type-2 problems. they are standardly solved! This presents a puzzle. How, given the statistical intractability of these type-2 cases, does nature turn the trick? One answer, which we do not pursue, is to suppose that evolution gifts us with exactly the right set of (...) recoding biases so as to reduce specific type-2 problems to (tractable) type-1 mappings. Such a heavy-duty nativism is no doubt sometimes plausible. But we believe there are other, more general mechanisms also at work. Such mechanisms provide general (not task-specific) strategies for managing problems of type-2 complexity. Several such mechanisms are investigated. At the heart of each is a fundamental ploy representational redescription language and culture – may themselves be viewed as adaptations enabling this representation/computation trade-off to be pursued on an even grander scale. (shrink)

During the last decades, many cognitive architectures (CAs) have been realized adopting different assumptions about the organization and the representation of their knowledge level. Some of them (e.g. SOAR [35]) adopt a classical symbolic approach, some (e.g. LEABRA[ 48]) are based on a purely connectionist model, while others (e.g. CLARION [59]) adopt a hybrid approach combining connectionist and symbolic representational levels. Additionally, some attempts (e.g. biSOAR) trying to extend the representational capacities of CAs by integrating diagrammatical representations and reasoning (...) are also available [34]. In this paper we propose a reflection on the role that Conceptual Spaces, a framework developed by Peter G¨ardenfors [24] more than fifteen years ago, can play in the current development of the Knowledge Level in Cognitive Systems and Architectures. In particular, we claim that Conceptual Spaces offer a lingua franca that allows to unify and generalize many aspects of the symbolic, sub-symbolic and diagrammatic approaches (by overcoming some of their typical problems) and to integrate them on a common ground. In doing so we extend and detail some of the arguments explored by G¨ardenfors [23] for defending the need of a conceptual, intermediate, representation level between the symbolic and the sub-symbolic one. In particular we focus on the advantages offered by Conceptual Spaces (w.r.t. symbolic and sub-symbolic approaches) in dealing with the problem of compositionality of representations based on typicality traits. Additionally, we argue that Conceptual Spaces could offer a unifying framework for interpreting many kinds of diagrammatic and analogical representations. As a consequence, their adoption could also favor the integration of diagrammatical representation and reasoning in CAs. (shrink)

Some regularities enjoy only an attenuated existence in a body of training data. These are regularities whose statistical visibility depends on some systematic recoding of the data. The space of possible recodings is, however, infinitely large – it is the space of applicable Turing machines. As a result, mappings that pivot on such attenuated regularities cannot, in general, be found by brute-force search. The class of problems that present such mappings we call the class of “type-2 problems.” Type-1 (...) problems, by contrast, present tractable problems of search insofar as the relevant regularities can be found by sampling the input data as originally coded. Type-2 problems, we suggest, present neither rare nor pathological cases. They are rife in biologically realistic settings and in domains ranging from simple animat behaviors to language acquisition. Not only are such problems rife – they are standardly solved! This presents a puzzle. How, given the statistical intractability of these type-2 cases, does nature turn the trick? One answer, which we do not pursue, is to suppose that evolution gifts us with exactly the right set of recoding biases so as to reduce specific type-2 problems to type-1 mappings. Such a heavy-duty nativism is no doubt sometimes plausible. But we believe there are other, more general mechanisms also at work. Such mechanisms provide general strategies for managing problems of type-2 complexity. Several such mechanisms are investigated. At the heart of each is a fundamental ploy – namely, the maximal exploitation of states of representation already achieved by prior, simpler learning so as to reduce the amount of subsequent computational search. Such exploitation both characterizes and helps make unitary sense of a diverse range of mechanisms. These include simple incremental learning, modular connectionism, and the developmental hypothesis of “representational redescription”. In addition, the most distinctive features of human cognition – language and culture – may themselves be viewed as adaptations enabling this representation/computation trade-off to be pursued on an even grander scale. (shrink)

b>: The problem of how physical systems, such as brains, come to represent themselves as subjects in an objective world is addressed. I develop an account of the requirements for this ability that draws on and refines work in a philosophical tradition that runs from Kant through Peter Strawson to Gareth Evans. The basic idea is that the ability to represent oneself as a subject in a world whose existence is independent of oneself involves the ability to represent (...) class='Hi'>space, and in particular, to represent oneself as one object among others in an objective spatial realm. In parallel, I provide an account of how this ability, and the mechanisms that support it, are realized neurobiologically. This aspect of the article draws on, and refines, work done in the neurobiology and psychology of egocentric and allocentric spatial representation. (shrink)

We illustrate the application of the conceptual analysis (CA) method outlined in Part I by the example of quantum mechanics. In the present part the Hilbert space structure of conventional quantum mechanics is deduced as a consequence of postulates specifying further idealized concepts. A critical discussion of the idealizations of quantum mechanics is proposed. Quantum mechanics is characterized as a “statistically complete” theory and a simple and elegant formal recipe for the construction of the fundamental mathematical apparatus of quantum (...) mechanics is formulated. Our analysis may also lead to a criticism of quantum mechanics as a “strongly idealized” theory. A critical analysis of the fundamental structure of quantum mechanics seems an indispensable and natural starting point for the construction of new theories. A major technical problem in a more general application of the CA method is the lack of mathematical representation theorems for more general algebraic structures. (shrink)

This paper starts with an intuitive notion of representational spaces, which is intended to provide an improved version of Kuhn's concept of paradigms. It then proceeds to study the following topics in terms of this new notion: incommensurability, paradigm change, explanation of anomalies, explanation of regularities, explanation of irregularities, and physical necessity. In the course of the investigation, "representational space" gets clarified and defined. It is envisaged that this new concept should throw light on many issues in the philosophy (...) of science. (shrink)

Spatial Representation presents original, specially written essays by leading psychologists and philosophers on a fascinating set of topics at the intersection of these two disciplines. They address such questions as these: Do the extraordinary navigational abilities of birds mean that these birds have the same kind of grip on the idea of a spatial world as we do? Is there a difference between the way sighted and blind subjects represent the world 'out there'? Does the study of brain-injured subjects, (...) such as 'blind seers', tell us anything about the working of normal spatial consciousness? -/- The essays are arranged into five sections, each of which reflects a central area of research into spatial cognition, and opens with a short introduction by the editors, designed to facilitate cross-disciplinary reading. The volume as a whole offers a rich and compelling expression of the view that to advance our understanding of the way we represent the external world it is necessary to draw on both philosophical and psychological approaches. (shrink)

We look at bimodal logics interpreted by cartesian products of topological spaces and discuss the validity of certain bimodal formulae in products of so-called cardinal spaces. This solves an open problem of van Benthem et al.

We investigate ideals of the form {A⊆ω: Σn∈Axnis unconditionally convergent} where n∈ωis a sequence in a Polish group or in a Banach space. If an ideal onωcan be seen in this form for some sequence inX, then we say that it is representable inX.After numerous examples we show the following theorems: An ideal is representable in a Polish Abelian group iff it is an analytic P-ideal. An ideal is representable in a Banach space iff it is a nonpathological (...) analytic P-ideal.We focus on the family of ideals representable inc0. We characterize this property via the defining sequence of measures. We prove that the trace of the null ideal, Farah’s ideal, and Tsirelson ideals are not representable inc0, and that a tallFσP-ideal is representable inc0iff it is a summable ideal. Also, we provide an example of a peculiar ideal which is representable inℓ1but not in ℝ.Finally, we summarize some open problems of this topic. (shrink)

Abstract:Unable to produce scales of one-to-one in places or spaces, people interact as they develop common representations, actions, and objects. In places, they can point to connect representations and objects. Places may be small spaces, but spaces can contain places and even more variations. Unable to name each variation, actors and analysts create abstractions. They might use a part to represent the whole, with miles of track for railroads, kilowatt-hours for electric power, and barrels of oil for energy. Some treat (...) their representations as hegemonic, even true. However, dealing with social and material variations, actors and analysts represent, build, maintain, and repair combinations of cultures, economies, governments, and materials. Thus, railroads and power systems, like other technologies and infrastructures, do not design, build, or repair themselves. To resolve problems, people with relevant skillsets interact in places to create common representations, actions, and objects tying themselves and others together in spaces. (shrink)

Most historians of science eagerly acknowledge that the early modern period witnessed a shift from a prevailing Aristotelian, spherical, centered conception of space to a prevailing Cartesian, rectilinear, oriented spatial framework. Indeed, this shift underlay many of the important advances for which the period is celebrated. However, historians have failed to engage the general conceptual shift, focusing instead on the particular explanatory developments that resulted. This historical lacuna can be attributed to a historiographical problem: the lack of an (...) adequate unit of analysis by which to investigate the conceptual change. Here, a philosophical argument is made for representations of space as an appropriate category of historical investigation, and methods of textual interrogation are suggested to this end. Finally, two examples, taken from Aristotle and Newton, demonstrate the feasibility and importance of this project. (shrink)

One important role of belief systems is to allow us to represent information about a certain domain of inquiry. This paper presents a formal framework to accommodate such information representation. Three cognitive models to represent information are discussed: conceptual spaces, state-spaces, and the problem spaces familiar from artificial intelligence. After indicating their weakness to deal with partial information, it is argued that an alternative, formulated in terms of partial structures, can be provided which not only captures the positive (...) features of these models, but also accommodates the partiality of information ubiquitous in science and mathematics. (shrink)

An early, very preliminary edition of this book was circulated in 1962 under the title Set-theoretical Structures in Science. There are many reasons for maintaining that such structures play a role in the philosophy of science. Perhaps the best is that they provide the right setting for investigating problems of representation and invariance in any systematic part of science, past or present. Examples are easy to cite. Sophisticated analysis of the nature of representation in perception is to be (...) found already in Plato and Aristotle. One of the great intellectual triumphs of the nineteenth century was the mechanical explanation of such familiar concepts as temperature and pressure by their representation in terms of the motion of particles. A more disturbing change of viewpoint was the realization at the beginning of the twentieth century that the separate invariant properties of space and time must be replaced by the space-time invariants of Einstein's special relativity. Another example, the focus of the longest chapter in this book, is controversy extending over several centuries on the proper representation of probability. The six major positions on this question are critically examined. Topics covered in other chapters include an unusually detailed treatment of theoretical and experimental work on visual space, the two senses of invariance represented by weak and strong reversibility of causal processes, and the representation of hidden variables in quantum mechanics. The final chapter concentrates on different kinds of representations of language, concluding with some empirical results on brain-wave representations of words and sentences. (shrink)

The framework of elementary probabilistic operations (EPO) explains the structure of elementary probabilistic reasoning tasks as well as people’s performance on these tasks. The framework comprises three components: (a) Three types of probabilities: joint, marginal, and conditional probabilities; (b) three elementary probabilistic operations: combination, marginalization, and conditioning, and (c) quantitative inference schemas implementing the EPO. The formal part of the EPO framework is a computational level theory that provides a problem space representation and a classification of elementary (...) probabilistic problems based on computational requirements for solving a problem. According to the EPO framework, current methods for improving probabilistic reasoning are of two kinds: First, reduction of Bayesian problems to a type of probabilistic problems requiring less conceptual and procedural competencies. Second, enhancing people’s utilization competence by fostering the application of quantitative inference schemas. The approach suggests new applications, including the teaching of probabilistic reasoning, using analogical problem solving in probabilistic reasoning, and new methods for analyzing errors in probabilistic problem solving. (shrink)

We consider countable so‐called rich subsemigroups of ; each such semigroup T gives a variety CPEAT that is axiomatizable by a finite schema of equations taken in a countable subsignature of that of ω‐dimensional cylindric‐polyadic algebras with equality where substitutions are restricted to maps in T. It is shown that for any such T, if and only if is representable as a concrete set algebra of ω‐ary relations. The operations in the signature are set‐theoretically interpreted like in polyadic equality set (...) algebras, but such operations are relativized to a union of cartesian spaces that are not necessarily disjoint. This is a form of guarding semantics. We show that CPEAT is canonical and atom‐canonical. Imposing an extra condition on T, we prove that atomic algebras in CPEAT are completely representable and that CPEAT has the super amalgamation property. If T is rich and finitely represented, it is shown that CPEAT is term definitionally equivalent to a finitely axiomatizable Sahlqvist variety. Such semigroups exist. This can be regarded as a solution to the central finitizability problem in algebraic logic for first order logic with equality if we do not insist on full fledged commutativity of quantifiers. The finite dimensional case is approached from the view point of guarded and clique guarded (relativized) semantics of fragments of first order logic using finitely many variables. Both positive and negative results are presented. (shrink)

We consider a representation of the state reduction which depends neither on its reality nor on the details of when and how it emerges. Then by means of the representation we find necessary conditions, even if not the sufficient ones, for a decomposition of the state vector space to be a solution to the basis problem. The conditions are that the decomposition should be Lorentz invariant and orthogonal and that the associated projections should be continuous. They (...) are shown to be able to determine a decomposition in each of a few examples considered if the other circumstances are taken into account together. (shrink)

In the course of daily life we solve problems often enough that there is a special term to characterize the activity and the right to expect a scientific theory to explain its dynamics. The classical view in psychology is that to solve a problem a subject must frame it by creating an internal representation of the problem‘s structure, usually called a problem space. This space is an internally generable representation that is mathematically identical (...) to a graph structure with nodes and links. The nodes can be annotated with useful information, and the whole representation can be distributed over internal and external structures such as symbolic notations on paper or diagrams. If the representation is distributed across internal and external structures the subject must be able to keep track of activity in the distributed structure. Problem solving proceeds as the subject works from an initial state in this mentally supported space, actively construction possible solution paths, evaluating them and heuristically choosing the best. Control of this exploratory process is not well understood, as it is not always systematic, but various heuristic search algorithms have been proposed and some experimental support has been provided for them. (shrink)

Primate foraging can be construed as a set of interconnected problems that include finding food, selecting efficient travel routes, anticipating the positions of moving prey, and manipulating, and occasionally, extracting food items using tools. The evidence reviewed in this paper strongly suggests that both monkeys and apes use three types of representation (i.e., static, dynamic, and relational) to solve various problems. Static representations involve recalling certain features of the environment, dynamic representations involve imagining changes in the trajectories of moving (...) objects, and relational representations involve encoding the properties of objects in relation to other objects. Contrary to previous claims, no clear differences were found between the representational skills of monkeys and apes. Current evidence also suggests that primates may be better at representing general compared to specific problem features. Finally, we have characterized the domains of space and objects as complementary and indicated future lines of research in these domains. (shrink)

In the course of daily life we solve problems often enough that there is a special term to characterize the activity and the right to expect a scientific theory to explain its dynamics. The classical view in psychology is that to solve a problem a subject must frame it by creating an internal representation of the problem’s structure, usually called a problem space. This space is an internally generable representation that is mathematically identical (...) to a graph structure with nodes and links. The nodes can be annotated with useful information, and the whole representation can be distributed over internal and external structures such as symbolic notations on paper or diagrams. If the representation is distributed across internal and external structures the subject must be able to keep track of activity in the distributed structure. Problem solving proceeds as the subject works from an initial state in mentally supported space, actively constructing possible solution paths, evaluating them and heuristically choosing the best. Control of this exploratory process is not well understood, as it is not always systematic, but various heuristic search algorithms have been proposed and some experimental support has been provided for them. (shrink)

Hermann Weyl was one of the early contributors to the mathematics of general relativity. This article argues that in 1929, for the formulation of a general relativistic framework of the Dirac equation, he both abolished and preserved in modified form the conceptual perspective that he had developed earlier in his “analysis of the problem of space.” The ideas of infinitesimal congruence from the early 1920s were aufgehoben in the general relativistic framework for the Dirac equation. He preserved the (...) central idea of gauge as a “purely infinitesimal” aspect of symmetries in a group extension schema. With respect to methodology, however, Weyl gave up his earlier preferences for relatively a-priori arguments and tried to incorporate as much empiricism as he could. This signified a clearly expressed empirical turn for him. Moreover, in this step he emphasized that the mathematical objects used for the representation of matter structures stood at the center of the construction, rather than interaction fields which, in the early 1920s, he had considered as more or less derivable from geometrico-philosophical considerations. (shrink)

In this paper, it will be argued that each time a procedure in a representational system is functioning adequately and automatically, the child steps up to a metaprocedural level and considers the procedure as a unit in its own right. Data will be drawn from microdevelopment in children's creation of external memory devices (i.e., changes in representation of a spatial task during a one hour's session) as well as from macrodevelopment in language acquisition (i.e., changes occurring over age in (...) a study of nominal determiners, pronouns and gender markers). These two aspects of development of representational systems are examined with respect to the spontaneous passage from concrete reference to abstract systems, from idiosyncratic to conventional notations, from redundant to economic marking, from unifunctional to plurifunctional meaningful units. These micro‐ and manodevelopmental changes take place in a spontaneous fashion despite the fact that the child's earlier system is adequate for successfully conveying the information required. A discussion will be made of the way in which children tend to indicate externally, by unifunctional markers, information which was implicit in their earlier use of an adult‐like system. They appear to be striving for the delicate balance between encoding and decoding efforts, ultimately leading to a felicitous representational system. Metaprocedural behavior is hypothesized to act as a control mechanism in the striving for this delicate balance of information content and information processing effort. It also appears to act as a control mechanism for the interprocedural organizotion of what was previously a plethora of juxtaposed, unconnected procedures yielding superficially analogous behavior. The stepping‐up to a metaprocedural level seems to occur each time the child has a handle on his currently functioning, goal‐oriented system. This enables the child to treat the system, or procedures which are part thereof, as units functioning in their own right. Thus there appears to be a deep‐rooted psychological tendency for a representational tool which functions well procedurally to become subsequently a problem‐space per se. (shrink)

We study computability of the abstract linear Cauchy problem equation image)where A is a linear operator, possibly unbounded, on a Banach space X. We give necessary and sufficient conditions for A such that the solution operator K: x ↦ u of the problem is computable. For studying computability we use the representation approach to computable analysis developed by Weihrauch and others. This approach is consistent with the model used by Pour-El/Richards.

The representation of mathematical objects in terms of (more) basic ones is part and parcel of (the foundations of) mathematics. In the usual foundations of mathematics, namely, ZFC set theory, all mathematical objects are represented by sets, while ordinary, namely, non–set theoretic, mathematics is represented in the more parsimonious language of second-order arithmetic. This paper deals with the latter representation for the rather basic case of continuous functions on the reals and Baire space. We show that the (...) logical strength of basic theorems named after Tietze, Heine, and Weierstrass changes significantly upon the replacement of “second-order representations” by “third-order functions.” We discuss the implications and connections to the reverse mathematics program and its foundational claims regarding predicativist mathematics and Hilbert’s program for the foundations of mathematics. Finally, we identify the problem caused by representations of continuous functions and formulate a criterion to avoid problematic codings within the bigger picture of representations. (shrink)

In a recent paper Griffiths [38] has argued, based on the consistent histories interpretation, that Hilbert space quantum mechanics is noncontextual. According to Griffiths the problem of contextuality disappears if the apparatus is “designed and operated by a competent experimentalist” and we accept the Single Framework Rule. We will argue from a representational realist stance that the conclusion is incorrect due to the misleading understanding provided by Griffiths to the meaning of quantum contextuality and its relation to physical (...) reality and measurements. We will discuss how the quite general incomprehension of contextuality has its origin in the ''objective-subjective omelette'' created by Heisenberg and Bohr. We will argue that in order to unscramble the omelette we need to disentangle, firstly, representational realism from naive realism, secondly, ontology from epistemology, and thirdly, the different interpretational problems of QM. In this respect, we will analyze what should be considered as Meaningful Physical Statements within a theory and will argue that Counterfactual Reasoning -considered by Griffiths as ''tricky''- must be accepted as a necessary condition for any representational realist interpretation of QM. Finally we discuss what should be considered as a problem in QM from a representational realist perspective. (shrink)

The concept of representation has been a key element in the scientific study of mental processes, ever since such studies commenced. However, usage of the term has been all but too liberal—if one were to adhere to common use it remains unclear if there are examples of physical systems which cannot be construed in terms of representation. The problem is considered afresh, taking as the starting point the notion of activity spaces—spaces of spatiotemporal events produced by dynamical (...) systems. It is argued that representation can be analyzed in terms of the geometrical and topological properties of such spaces. Several attributes and processes associated with conceptual domains, such as logical structure, generalization and learning are considered, and given analogues in structural facets of activity spaces, as are misrepresentation and states of arousal. Based on this analysis, representational systems are defined, as is a key concept associated with such systems, the notion of representational capacity. According to the proposed theory, rather than being an all or none phenomenon, representation is in fact a matter of degree—that is can be associated with measurable quantities, as is behooving of a putative naturalistic construct. (shrink)

A semantics of pictorial representation should provide an account of how pictorial signs are associated with the contents they express. Unlike the familiar semantics of spoken languages, this problem has a distinctively spatial cast for depiction. Pictures themselves are two-dimensional artifacts, and their contents take the form of pictorial spaces, perspectival arrangements of objects and properties in three dimensions. A basic challenge is to explain how pictures are associated with the particular pictorial spaces they express. Inspiration here comes (...) from recent proposals that analyze depiction in terms of geometrical projection. In this essay, I will argue that, for a central class of pictures, the projection-based theory of depiction provides the best explanation for how pictures express pictorial spaces, while rival perceptual and resemblance theories fall short. Since the composition of pictorial space is itself the basis for all other aspects of pictorial content, the proposal provides a natural foundation for further pictorial semantics. (shrink)

A problem is a situation in which an agent seeks to attain a given goal without knowing how to achieve it. Human problem solving is typically studied as a search in a problem space composed of states (information about the environment) and operators (to move between states). A problem such as playing a game of chess has possible states, and a traveling salesperson problem with as little as 82 cities already has more than different (...) tours (similar to chess). Biological neurons are slower than the digital switches in computers. An exhaustive search of the problem space exceeds the capacity of current computers for most interesting problems, and it is fairly clear that humans cannot in their lifetime exhaustively search even small fractions of these problem spaces. Yet, humans play chess and solve logistical problems of similar complexity on a daily basis. Even for simple problems humans do not typically engage in exploring even a small fraction of the problem space. This begs the question: How do humans solve problems on a daily basis in a fast and efficient way? Recent work suggests that humans build a problem representation and solve the represented problem—not the problem that is out there. The problem representation that is built and the process used to solve it are constrained by limits of cognitive capacity and a cost–benefit analysis discounting effort and reward. In this article, we argue that better understanding the way humans represent and solve problems using heuristics can help inform how simpler algorithms and representations can be used in artificial intelligence to lower computational complexity, reduce computation time, and facilitate real-time computation in complex problem solving. (shrink)

A set theory model of reality, representation and language based on the relation of completeness and incompleteness is explored. The problem of completeness of mathematics is linked to its counterpart in quantum mechanics. That model includes two Peano arithmetics or Turing machines independent of each other. The complex Hilbert space underlying quantum mechanics as the base of its mathematical formalism is interpreted as a generalization of Peano arithmetic: It is a doubled infinite set of doubled Peano arithmetics (...) having a remarkable symmetry to the axiom of choice. The quantity of information is interpreted as the number of elementary choices (bits). Quantum information is seen as the generalization of information to infinite sets or series. The equivalence of that model to a quantum computer is demonstrated. The condition for the Turing machines to be independent of each other is reduced to the state of Nash equilibrium between them. Two relative models of language as game in the sense of game theory and as ontology of metaphors (all mappings, which are not one-to-one, i.e. not representations of reality in a formal sense) are deduced. (shrink)

In this paper, it will be argued that each time a procedure in a representational system is functioning adequately and automatically, the child steps up to a metaprocedural level and considers the procedure as a unit in its own right. Data will be drawn from microdevelopment in children's creation of external memory devices (i.e., changes in representation of a spatial task during a one hour's session) as well as from macrodevelopment in language acquisition (i.e., changes occurring over age in (...) a study of nominal determiners, pronouns and gender markers). These two aspects of development of representational systems are examined with respect to the spontaneous passage from concrete reference to abstract systems, from idiosyncratic to conventional notations, from redundant to economic marking, from unifunctional to plurifunctional meaningful units. These micro‐ and manodevelopmental changes take place in a spontaneous fashion despite the fact that the child's earlier system is adequate for successfully conveying the information required. A discussion will be made of the way in which children tend to indicate externally, by unifunctional markers, information which was implicit in their earlier use of an adult‐like system. They appear to be striving for the delicate balance between encoding and decoding efforts, ultimately leading to a felicitous representational system. Metaprocedural behavior is hypothesized to act as a control mechanism in the striving for this delicate balance of information content and information processing effort. It also appears to act as a control mechanism for the interprocedural organizotion of what was previously a plethora of juxtaposed, unconnected procedures yielding superficially analogous behavior. The stepping‐up to a metaprocedural level seems to occur each time the child has a handle on his currently functioning, goal‐oriented system. This enables the child to treat the system, or procedures which are part thereof, as units functioning in their own right. Thus there appears to be a deep‐rooted psychological tendency for a representational tool which functions well procedurally to become subsequently a problem‐space per se. (shrink)

In the movie The Gold Rush Charlie Chaplin and his friend are stranded in a log cabin in the middle of winter while a blizzard rages. The cabin is isolated, and they have a very big problem – there is nothing to eat. They pace around wondering what to do. Charlie's friend starts to see Charlie as a chicken, and he tries to kill him. He chases Charlie around the cabin many times. Eventually they hit upon the idea of (...) boiling an old boot and eating it for dinner. With great delicacy they sit at the table and eat the boot as if it were a gourmet meal. They solved the problem of having nothing to eat. While their solution to the problem did not result in a culinary feast, this example reveals two crucial features of problem solving. First, a problem exists when a goal must be achieved and the solution is not immediately obvious. Second, problem solving often involves attempting different ways to solve the problem. Put more formally, a problem has four components. First, there is an initial state. This is the person's state of knowledge at the start of a problem. Second, there is the goal state: the goal that the person wishes to achieve. Third are the actions or operations that the problem‐solver can use to get to the goal state. Fourth is the task environment that the solver is working in. The task environment consists of the features of the physical environment that can either directly or indirectly constrain or suggest different ways of solving a problem. I will sketch out the main currents of thinking in research in this area, beginning by reviewing the history of research on problem solving and then focusing on a number of important issues in problem‐solving research. Finally, I will give an overview of some recent developments. (shrink)

We outline three challenges involved in designing external representations that promote sustainable use of natural resources. First, the task environment of sustainable resource-use is highly unstructured, and involves many uncoordinated and asynchronous actions. Following from this complex nature of the task environment, more task constraints and task interactions are involved in designing representations promoting sustainability, compared to representations that seek to make tasks easier in structured task environments, such as aircraft cockpits and control rooms. Second, external representations promoting sustainable resource-use (...) need to motivate people to make decisions that sustain resources, and persist with this behavior, even though alternate behaviors are easier and commonplace. Third, external representations promoting sustainability also need to lower the cognitive load involved in sustainability decisions. This three-tiered function (meeting complex task constraints, providing motivation, lowering cognitive load) makes such representations challenging to design. However, some early prototype designs promoting sustainable resource-use have appeared recently, primarily addressing electricity use. Analyzing these digital prototypes, we outline three design principles they share, and show how these seek to address the complexities of the sustainability problem-space (complexity of task environments, goals, and representations). We then argue that at least two further cognitive scaffolds are required for effectively promoting sustainable resource use (action scripts, deconstruction). We close with some of the limitations of this approach to promoting sustainability, and outline future work. (shrink)

Within the framework of this article, the authors analyze the cycle of engravings "Apocalypse" by Albrecht Durer in the context of the categories of space and time that have developed in the history and philosophy of culture of the Middle Ages and Renaissance. The metaphysical essence of time, defined by Christian eschatology, found its vivid embodiment in the activities of many figures of artistic culture of that era. Apocalyptic moods, which largely determine the consciousness of people of the Reformation (...) era, were also inherent in A. Durer. The cycle of engravings "Apocalypse" in many ways should be considered as the quintessence of these moods and a reflection of the image of death that develops in this era. Space also acquires metaphysical characteristics, becomes heterogeneous, emotionally colored. The authors conclude that in the "Apocalypse" there is a mixed (and therefore very peculiar) interpretation of the category of time and space. It clearly shows the transition of the space-time representations of the Middle Ages to the ideas of time and space inherent in the New Time. It is also possible that we have before us one of the variants of the Renaissance mastery of the categories of time and space in the art of Europe. Attention is drawn to its openness and conscious demonstrativeness, because two artistic systems interact with each other on an equal footing; their stylistic potential and expressive means are used entirely, with full dedication and as a result, with maximum impressive force. (shrink)

Closure spaces are generalizations of topological spaces, in which the Intersection of two open sets need not be open. The considered logic is related to closure spaces just as the standard logic S4 to topological ones. After describing basic properties of the logic we consider problems of representation of Lindenbaum algebras with some uncountable sets of infinite joins and meets, a notion of equality and a meaning of quantifiers. Results are extended onto the standard logic S4 and they are (...) valid also for some other standard modal logics. (shrink)

Concept representation is still an open problem in the field of ontology engineering and, more generally, of knowledge representation. In particular, the issue of representing “non classical” concepts, i.e. concepts that cannot be defined in terms of necessary and sufficient conditions, remains unresolved. In this paper we review empirical evidence from cognitive psychology, according to which concept representation is not a unitary phenomenon. On this basis, we sketch some proposals for concept representation, taking into account (...) suggestions from psychological research. In particular, it seems that human beings employ both prototype-based and exemplar-based representations in order to represent non classical concepts. We suggest that a similar, hybrid prototype-exemplar based approach could also prove useful in the field of knowledge representation technology. Finally, we propose conceptual spaces as a suitable framework for developing some aspects of this proposal. (shrink)

It is no surprise that the phenomenal qualities of our sensory experience pose recalcitrant philosophical problems for a physical materialist metaphysics. The colors of flowers as we experience them by sight, the taste of a ripe peach, and the smell of fresh-cut grass are undeniably part of the experienced world; yet in their phenomenal mode, they do not seem well-placed in the physicist's world of particles and energy fields. It seems, prima facie, that the metaphysical programs found in earlier science (...) and philosophy were better suited to accommodate these qualities: in the hylomorphic world of the Aristotelians, colors were "real qualities" existing as such in flowers; in the dualistic world of Descartes, colors were displaced from things like flowers to the immaterial mind of the perceiver. The dissertation argues that this intuition about our philosophical heritage is both philosophically confused and historically inaccurate. It betrays a misconception about phenomenal qualities and the problems they pose which results from a failure to distinguish phenomenal qualities from a special subset of sensible qualities that we have come to call "secondary" qualities. Disentangled from the primary-secondary quality distinction, phenomenal qualities include all sensible qualities insofar as they form the experiential contents of our sensory perceptions. So considered, phenomenal qualities invite difficult questions about the representational nature and ontological status of our sensory experience even within the Aristotelian and Cartesian metaphysics. By examining Descartes' and the Aristotelians' theories of sense perception with a focus on these questions we achieve a better understanding of "the problem" of phenomenal qualities, better interpretations of these historical theories of sense perception, and suggestions for reshaping the problem space within which we think philosophically about phenomenal qualities today. (shrink)

We present an account of semantics that is not construed as a mapping of language to the world but rather as a mapping between individual meaning spaces. The meanings of linguistic entities are established via a “meeting of minds.” The concepts in the minds of communicating individuals are modeled as convex regions in conceptual spaces. We outline a mathematical framework, based on fixpoints in continuous mappings between conceptual spaces, that can be used to model such a semantics. If concepts are (...) convex, it will in general be possible for interactors to agree on joint meaning even if they start out from different representational spaces. Language is discrete, while mental representations tend to be continuous—posing a seeming paradox. We show that the convexity assumption allows us to address this problem. Using examples, we further show that our approach helps explain the semantic processes involved in the composition of expressions. (shrink)

Knowledge representation is a central issue in a number of areas, but few attempts are usually made to bridge different approaches accross different fields. As a contribution in this direction, in this paper I focus on one such approach, the theory of conceptual spaces developed within cognitive science, and explore its potential applications in the fields of philosophy of science and formal epistemology. My case-study is provided by the theory of truthlikeness, construed as closeness to “the whole truth” about (...) a given domain, as described in the underlying language. I show how modeling propositions and their relations within a conceptual space has interesting implications for two issues in truthlikeness theory: the so called problem of language dependence, and that of measure sensitivity. I conclude by pointing at some open issues arising from the application of conceptual spaces to the analysis of philosophical problems. (shrink)

The aim of the paper is to assess the relative merits of two formal representations of structure, namely, set theory and category theory. The purpose is to articulate ontic structural realism. In turn, this will facilitate a discussion on the strengths and weaknesses of both concepts and will lead to a proposal for a pragmatics-based approach to the question of the choice of an appropriate framework. First, we present a case study from contemporary science—a comparison of the formulation of quantum (...) mechanics in the language of Hilbert spaces and abstract C* algebras. It is then shown how the method of structural representation can be determined based on the pragmatics of goal-oriented research, not a dogmatic choice. We investigate a hypothesis stating that the use of the interplay between the powers of abstraction and detail of different representational methods results in adopting a pluralistic, as opposed to standard, unificatory, perspective on the role of structural representation in OSR. (shrink)

This essay presents a philosophical and computational theory of the representation of de re, de dicto, nested, and quasi-indexical belief reports expressed in natural language. The propositional Semantic Network Processing System (SNePS) is used for representing and reasoning about these reports. In particular, quasi-indicators (indexical expressions occurring in intentional contexts and representing uses of indicators by another speaker) pose problems for natural-language representation and reasoning systems, because--unlike pure indicators--they cannot be replaced by coreferential NPs without changing the meaning (...) of the embedding sentence. Therefore, the referent of the quasi-indicator must be represented in such a way that no invalid coreferential claims are entailed. The importance of quasi-indicators is discussed, and it is shown that all four of the above categories of belief reports can be handled by a single representational technique using belief spaces containing intensional entities. Inference rules and belief-revision techniques for the system are also examined. (shrink)

A Satisfactory discussion in depth of all the philosophical problems that could be raised concerning musical representation would require much more space as well as more ability than I have at my disposal. Nobody should believe, or believe that I believe, that what follows is more than a rather sketchy examination of a few central issues.

Intelligent systems are faced with the problem of securing a principled (ideally, veridical) relationship between the world and its internal representation. I propose a unified approach to visual representation, addressing both the needs of superordinate and basic-level categorization and of identification of specific instances of familiar categories. According to the proposed theory, a shape is represented by its similarity to a number of reference shapes, measured in a high-dimensional space of elementary features. This amounts to embedding (...) the stimulus in a low-dimensional proximal shape space. That space turns out to support representation of distal shape similarities which is veridical in the sense of Shepard's (1968) notion of second-order isomorphism (i.e., correspondence between distal and proximal similarities among shapes, rather than between distal shapes and their proximal representations). Furthermore, a general expression for similarity between two stimuli, based on comparisons to reference shapes, can be used to derive models of perceived similarity ranging from continuous, symmetric, and hierarchical, as in the multidimensional scaling models (Shepard, 1980), to discrete and non-hierarchical, as in the general contrast models (Tversky, 1977; Shepard and Arabie, 1979). (shrink)

In this paper we develop a general representation theory for MV-algebras. We furnish the appropriate categorical background to study this problem. Our guide line is the theory of classifying topoi of coherent extensions of universal algebra theories. Our main result corresponds, in the case of MV-algebras and MV-chains, to the representation of commutative rings with unit as rings of global sections of sheaves of local rings. We prove that any MV-algebra is isomorphic to the MV-algebra of all (...) global sections of a sheaf of MV-chains on a compact topological space. This result is intimately related to McNaughton’s theorem, and we explain why our representation theorem can be viewed as a vast generalization of McNaughton’s theorem. In spite of the language used in this abstract, we have written this paper in the hope that it can be read by experts in MV-algebras but not in sheaf theory, and conversely. (shrink)

A Satisfactory discussion in depth of all the philosophical problems that could be raised concerning musical representation would require much more space as well as more ability than I have at my disposal. Nobody should believe, or believe that I believe, that what follows is more than a rather sketchy examination of a few central issues.

The representational epistemic approach to the design of visual displays and notation systems advocates encoding the fundamental conceptual structure of a knowledge domain directly in the structure of a representational system. It is claimed that representations so designed will benefit from greater semantic transparency, which enhances comprehension and ease of learning, and plastic generativity, which makes the meaningful manipulation of the representation easier and less error prone. Epistemic principles for encoding fundamental conceptual structures directly in representational schemes are described. (...) The diagrammatic recodification of probability theory is undertaken to demonstrate how the fundamental conceptual structure of a knowledge domain can be analyzed, how the identified conceptual structure may be encoded in a representational system, and the cognitive benefits that follow. An experiment shows the new probability space diagrams are superior to the conventional approach for learning this conceptually challenging topic. (shrink)

This thesis has the following aims. First, to show that Deleuze can be situated clearly within the post-Kantian tradition. This is achieved through an analysis of the relations between Kant's transcendental idealism and Deleuze's transcendental empiricism. Second, to explore the criticisms of representational theories of difference which can be found in the work of Deleuze and Hegel. Representational theories are best understood as theories which rely on a logic which is governed by relations between entities which pre-exist those relations. Deleuze (...) argues that these logics presuppose the formal equivalent of a homogeneous space within which these relations can be construed. Hegel similarly understands representation as the utilisation of finite categories which rely on the fixity of the subject of predication. The third aim is to provide a rigorous explication of some of the key themes of Deleuzian ontology, particularly in relation to the problem of representation. This will involve looking at the logic of multiplicities, which attempts to provide a theory of difference that is non-oppositional. This logic will be clarified through a discussion of Deleuze's use of modern geometry, and his analysis of the foundations of the calculus. The fourth aim will be to contrast Deleuze's solution with that of Hegel, particularly with respect to their relationships to Kant and the calculus. This is achieved through the Deleuzian distinction between finite and infinite representation, the latter in Deleuze's view characterising the Hegelian attempt to bring the idea of transition into representation itself. Finally, having shown where Deleuze and Hegel differ in their respective projects, the thesis will explore whether either of these philosophies has the resources to provide a refutation of the other with reference to the dialectic of force and the understanding in the henomenology of Spirit, and the problem of the one and the many. (shrink)