The goals of the present study were to test whether (and which) basic numerical abilities can be improved with training and whether training effects transfer to improvement in children’s math achievement. The literature is mixed with evidence that does or does not substantiate the efficacy of training basic numerical ability. In the present study, we developed a child-friendly software named ‘123 Bakery’ which includes four training modules; non-symbolic numerosity comparison, non-symbolic numerosity estimation, approximate arithmetic and symbol-to-numerosity (...) mapping. Fifty-six first graders were randomly assigned to either the training or control group. The training group participated in 6 weeks of training (5 times a week, 30 minutes per day). All participants underwent pre- and post- training assessment of their basic numerical processing ability (including numerosity discrimination acuity, symbolic/non-symbolic magnitude estimation, approximate arithmetic, symbol-to-numerosity mapping), overall math achievement and intelligence, 6 weeks apart. The acuity for numerosity discrimination (approximate number sense acuity; ANS acuity) significantly improved after training, but this training effect did not transfer to improvement in symbolic, exact calculation, or any other math ability. We conclude that basic numerical cognition training leads to improvement in ANS acuity, but whether this effect transfers to symbolic math ability remains to be further tested. (shrink)

The number-line estimation task has become one of the most important methods in numerical cognition research. Originally applied as a direct measure of spatial number representation, it became also informative regarding various other aspects of number processing and associated strategies. However, most of this work and associated conclusions concerns processing numbers in a symbolic format, by school children and older subjects. Symbolic number system is formally taught and trained at school, and its basic mathematical properties can easily be transferred into (...) a spatial format of an oriented number line. This triggers the question on basic characteristics of number line estimation before children get fully familiar with the symbolic number system, i.e., when they mostly rely on approximate system for non-symbolic quantities. In our three studies, we examine therefore how preschool children estimate position of non-symbolic quantities on a line, and how this estimation is related to the developing symbolic number knowledge and cultural directionality. The children were tested with the Give-a-number task, then they performed a computerized number-line task. In Experiment 1, lines bounded with sets of 1 and 20 elements going left-to-right or right-to-left were used. Even in the least numerically competent group, the linear model better fit the estimates than the logarithmic or cyclic power models. The line direction was irrelevant. In Experiment 2, a 1–9 left-to-right oriented line was used. Advantage of linear model was found at group level, and variance of estimates correlated with tested numerosities. In Experiment 3, a position-to-number procedure again revealed the advantage of the linear model, although the strategy of selecting an option more similar to the closer end of the line was prevalent. The precision of estimation increased with the mastery of counting principles in all three experiments. These results contradict the hypothesis of the log-to-linear shift in development of basic numerical representation, rather supporting the linear model with scalar variance. However, the important question remains whether the number-line task captures the nature of the basic numerical representation, or rather the strategies of mapping that representation to an external space. (shrink)

Mature representations of number are built on a core system of numerical representation that connects to spatial representations in the form of a ‘mental number line’. The core number system is functional in early infancy, but little is known about the origins of the mapping of numbers onto space. Here we show that preverbal infants transfer the discrimination of an ordered series of numerosities to the discrimination of an ordered series of line lengths. Moreover, infants construct relationships between individual (...) numbers and line lengths that vary positively, but not between numbers and lengths that vary inversely. These findings provide evidence for an early developing predisposition to relate representations of numerical magnitude and spatial length. A central foundation of mathematics, science and technology therefore emerges prior to experience with language, symbol systems, or measurement devices. (shrink)

Five-year-old children categorized as skilled versus unskilled counters were given verbal estimation and number word comprehension tasks with numerosities 20 – 120. Skilled counters showed a linear relation between number words and nonsymbolic numerosities. Unskilled counters showed the same linear relation for smaller numbers to which they could count, but not for larger number words. Further tasks indicated that unskilled counters failed even to correctly order large number words differing by a 2 : 1 ratio, whereas they performed well on (...) this task with smaller numbers, and performed well on a nonsymbolic ordering task with the same numerosities. These findings provide evidence that large, approximate numerosity representations become linked to number words around the time that children learn to count to those words reliably. (shrink)

It is shown in ZF (without choice) that if there is a finite-to-one map P(X) → X, then X is finite.

Five-year-old children categorized as skilled versus unskilled counters were given verbal estimation and number word comprehension tasks with numerosities 20 – 120. Skilled counters showed a linear relation between number words and nonsymbolic numerosities. Unskilled counters showed the same linear relation for smaller numbers to which they could count, but not for larger number words. Further tasks indicated that unskilled counters failed even to correctly order large number words differing by a 2 : 1 ratio, whereas they performed well on (...) this task with smaller numbers, and performed well on a nonsymbolic ordering task with the same numerosities. These findings provide evidence that large, approximate numerosity representations become linked to number words around the time that children learn to count to those words reliably. (shrink)

ABSENCE is the claim that, if a symbol appears on a map, then absence of the symbol from some map coordinate signifies absence of the corresponding property from the corresponding location. This claim is highly intuitive and widely endorsed. And if it is true, then cartographic representation is strikingly different from linguistic representation. I argue, however, that ABSENCE is false of various maps and that we have no reason to believe it is true of any maps. The intuition (...) to the contrary results from mistaking what a map simply conveys for what it literally represents. (shrink)

Sound symbolism refers to non‐arbitrary mappings between the sounds of words and their meanings and is often studied by pairing auditory pseudowords such as “maluma” and “takete” with rounded and pointed visual shapes, respectively. However, it is unclear what auditory properties of pseudowords contribute to their perception as rounded or pointed. Here, we compared perceptual ratings of the roundedness/pointedness of large sets of pseudowords and shapes to their acoustic and visual properties using a novel application of representational similarity analysis (RSA). (...) Representational dissimilarity matrices (RDMs) of the auditory and visual ratings of roundedness/pointedness were significantly correlated crossmodally. The auditory perceptual RDM correlated significantly with RDMs of spectral tilt, the temporal fast Fourier transform (FFT), and the speech envelope. Conventional correlational analyses showed that ratings of pseudowords transitioned from rounded to pointed as vocal roughness (as measured by the harmonics‐to‐noise ratio, pulse number, fraction of unvoiced frames, mean autocorrelation, shimmer, and jitter) increased. The visual perceptual RDM correlated significantly with RDMs of global indices of visual shape (the simple matching coefficient, image silhouette, image outlines, and Jaccard distance). Crossmodally, the RDMs of the auditory spectral parameters correlated weakly but significantly with those of the global indices of visual shape. Our work establishes the utility of RSA for analysis of large stimulus sets and offers novel insights into the stimulus parameters underlying sound symbolism, showing that sound‐to‐shape mapping is driven by acoustic properties of pseudowords and suggesting audiovisual cross‐modal correspondence as a basis for language users' sensitivity to this type of sound symbolism. (shrink)

Human adults are thought to possess two dissociable systems to represent numbers: an approximate quantity system akin to a mental number line, and a verbal system capable of representing numbers exactly. Here, we study the interface between these two systems using an estimation task. Observers were asked to estimate the approximate numerosity of dot arrays. We show that, in the absence of calibration, estimates are largely inaccurate: responses increase monotonically with numerosity, but underestimate the actual numerosity. However, (...) insertion of a few inducer trials, in which participants are explicitly (and sometimes misleadingly) told that a given display contains 30 dots, is sufficient to calibrate their estimates on the whole range of stimuli. Based on these empirical results, we develop a model of the mapping between the numerical symbols and the representations of numerosity on the number line. (shrink)

This paper reports the framework, method and main findings of an analysis of cultural milieus in 4 European countries. The analysis is based on a questionnaire applied to a sample built through a two-step procedure of post-hoc random selection from a broader dataset based on an online survey. Responses to the questionnaire were subjected to multidimensional analysis-a combination of Multiple Correspondence Analysis and Cluster Analysis. We identified 5 symbolic universes, that correspond to basic, embodied, affect-laden, generalized worldviews. People in this (...) study see the world as either a) an ordered universe;b) a matter of interpersonal bond;c) a caring society;d) consisting of a niche of belongingness;e) a hostile place. These symbolic universes were also interpreted as semiotic capital: they reflect the capacity of a place to foster social and civic development. Moreover, the distribution of the symbolic universes, and therefore social and civic engagement, is demonstrated to be variable across the 4 countries in the analysis. Finally, we develop a retrospective reconstruction of the distribution of symbolic universes as well as the interplay between their current state and past, present and future socio-institutional scenarios. (shrink)

The number of elements in a small set of items is appraised in a fast and exact manner, a phenomenon called subitizing. In contrast, humans provide imprecise responses when comparing larger numerosities, with decreasing precision as the number of elements increases. Estimation is thought to rely on a dedicated system for the approximate representation of numerosity. While previous behavioral and neuroimaging studies associate subitizing to a domain-general system related to object tracking and identification, the nature of small numerosity (...) processing is still debated. We investigated the neural processing of numerosity across subitizing and estimation ranges by examining electrophysiological activity during the memory retention period in a delayed numerical match-to-sample task. We also assessed potential differences in the neural signature of numerical magnitude in a fully non-symbolic or cross-format comparison. In line with behavioral performance, we observed modulation of parietal-occipital neural activity as a function of numerosity that differed in two ranges, with distinctive neural signatures of small numerosities showing clear similarities with those observed in visuospatial working memory tasks. We also found differences in neural activity related to numerical information in anticipation of single vs. cross-format comparison, suggesting a top-down modulation of numerical processing. Finally, behavioral results revealed enhanced performance in the mixed-format conditions and a significant correlation between task performance and symbolic mathematical skills. Overall, we provide evidence for distinct mechanisms related to small and large numerosity and differences in numerical encoding based on task demands. (shrink)

ABSTRACTThe rapid expansion of primary education in Ethiopia has enabled most children to attend primary education—or at least to start schooling. This expansion, however, is largely “symbolic” rat...

While the 1970's have been called the environmental years, the 1990's could be seen as the genome years. As the challenge to map and to sequence the human genome mobilized the scientific community, risks and benefits of information and uses that would derive from this project have also raised ethical issues at the international level. The particular interest of the 1997 UNESCO Declaration relies on the fact that it emphasizes both the scientific importance of genetics and the appropriate reinforcement of (...) human rights in this area. It considers the human genome, at least symbolically, as the common heritage of humanity. (shrink)

A shorthand symbolism for the relational mapping of categories is introduced and developed on the basis of Nelson Goodman's structural methodology. Through a reconstruction of extensional isomorphism that Goodman introduces as a criterion for definitional accuracy, and a brief reminder of the argument structure behind his ‘new riddle of induction’, Goodman's radical ontological relativism is turned into a protological principle of what I call ‘domain constituting philosophy’. MSS is demonstrated with reference to Goodman's symbol theory, particularly his notion (...) of exemplification, as well as in a final structured comparison of Frege's ‘sense’ with Goodman's straightforward construction of ‘secondary extension’. (shrink)

To describe and analyze the unstable periodic orbits of the Rucklidge system, a so-called symbolic encoding method is introduced, which has been proven to be an efficient tool to explore the topological properties concealed in these periodic orbits. In this work, the unstable periodic orbits up to a certain topological length in the Rucklidge system are systematically investigated via a proposed variational method. The dynamics in the Rucklidge system are explored by using phase portrait analysis, Lyapunov exponents, and Poincaré first (...) return maps. Symbolic encodings of the periodic orbits with two and four letters based on the trajectory topology in the phase space are implemented under two sets of parameter values. Meanwhile, the bifurcations of the periodic orbits are explored, significantly improving the understanding of the dynamics of the Rucklidge system. The multiple-letter symbolic encoding method could also be applicable to other nonlinear dynamical systems. (shrink)

Several different versions of the theory of numerosities have been introduced in the literature. Here, we unify these approaches in a consistent frame through the notion of set of labels, relating numerosities with the Kiesler field of Euclidean numbers. This approach allows us to easily introduce, by means of numerosities, ordinals and their natural operations, as well as the Lebesgue measure as a counting measure on the reals.

Tagging is the phenomenon in which regions of a picture, map, or diagram are annotated with words or other symbols, to provide descriptive information about a depicted object. The interpretive principles that govern tagged images are not well understood, due in part to the difficulty of integrating pictorial and linguistic semantic rules. Rather than directly combining these rules, I propose to use the framework of perspectival feature maps as an intermediary representation of content, in which the outputs of pictorial and (...) linguistic interpretation may be assimilated. The result is a simple and compositional semantics for tagged images. (shrink)

We investigate computable subshifts and the connection with effective symbolic dynamics. It is shown that a decidable Π01 class P is a subshift if and only if there exists a computable function F mapping 2ℕ to 2ℕ such that P is the set of itineraries of elements of 2ℕ. Π01 subshifts are constructed in 2ℕ and in 2ℤ which have no computable elements. We also consider the symbolic dynamics of maps on the unit interval.

It is common to distinguish two great families of representation. Symbolic representations include logical and mathematical symbols, words, and complex linguistic expressions. Iconic representations include dials, diagrams, maps, pictures, 3-dimensional models, and depictive gestures. This essay describes and motivates a new way of distinguishing iconic from symbolic representation. It locates the difference not in the signs themselves, nor in the contents they express, but in the semantic rules by which signs are associated with contents. The two kinds of rule have (...) divergent forms, occupying opposite poles on a spectrum of naturalness. Symbolic rules are composed entirely of primitive juxtapositions of sign types with contents, while iconic rules determine contents entirely by uniform natural relations with sign types. This distinction is marked explicitly in the formal semantics of familiar sign systems, both for atomic first-order representations, like words, pixel colors, and dials, and for complex second-order representations, like sentences, diagrams, and pictures. (shrink)

The proposed model consists of 1) a Phonological Planning Component to plan the symbolic and relational goals for an utterance, 2) a Phonetic Planning Component to plan the quantitative details of the acoustic goals and how they will be achieved articulatorily, and 3) a Motor-Sensory Implementation Component to ensure that the goals are achieved on time. The temporal characteristics specified in the Phonetic Planning Component include durations between acoustic landmarks, as well as parameters of Lee’s TauG-Guidance equation, which determine how (...) a movement evolves over time. Movement coordination is assumed to be based on movement endpoints (rather than onsets) and accomplished via Tau-coupling (Lee 1998). Both the Phonetic Planning and Motor-Sensory Implementation components are assumed to make use of general-purpose, phonology-extrinsic, timekeeping mechanisms to represent, specify, and track time. Supporting evidence includes greater timing precision at movement endpoints compared to other parts of movements, suggesting the separate control of the timing of movement endpoints compared to other parts of movement. This evidence challenges models in which all parts of a movement trajectory are controlled by the same equation of motion, but supports models in which a) each abstract, symbolic phonological representation maps onto spatial and temporal characteristics of the part(s) of movement most closely related to the goal of producing a planned set of acoustic cues to signal the phonological contrast (often the endpoint), b) movements are coordinated primarily based on the goal-related part of movement, and c) speakers give priority to the accurate implementation of the part(s) of movement most closely related to the phonological goals. Further support for phonology-extrinsic timing comes from three types of evidence suggesting that surface durations are represented during speech production. Finally, phonology-extrinsic timing is supported by greater timing variability for repetitions of longer intervals, assumed to be due to noise in a general-purpose (and phonology-extrinsic) timekeeping process. The evidence appears to be incompatible with models that have a unified Phonology/Phonetics Component, with models which do not represent the surface timing of phonetic events, nor with models in which timing in speech production is not represented, specified and tracked by a general-purpose timekeeping mechanisms. (shrink)

Fodor advocates a view of cognitive processes as computations defined over the language of thought (or Mentalese). Even among those who endorse Mentalese, considerable controversy surrounds its representational format. What semantically relevant structure should scientific psychology attribute to Mentalese symbols? Researchers commonly emphasize logical structure, akin to that displayed by predicate calculus sentences. To counteract this tendency, I discuss computational models of navigation drawn from probabilistic robotics. These models involve computations defined over cognitive maps, which have geometric rather than logical (...) structure. They thereby demonstrate the possibility of rational cognitive processes in an exclusively non-logical representational medium. Furthermore, they offer much promise for the empirical study of animal navigation. (shrink)

This study employs an experiment investigating cognitive mapping of fifth-grade children living in a remote village environment, wherein characteristics of the landscape included paths, landmarks, nodes, edges, and districts. Two aspects of analysis were salient in this study. First, important landscape characteristics and their frequency of appearance in the cognitive maps were tabulated and illustrated as a layout map. Second, inaccurate cognitive maps were structurally analyzed to account for any incompleteness, distortions, and augmentation of actual environments found in some (...) map samples. Focus on gender differences in children’s environmental cognition in terms of symbolic representation skills utilized in cognitive mapping is of special interest in this case study. Results confirm Piaget’s theory that older children, aged ten or more, begin to use projective and Euclidean concepts. Furthermore, boys used a greater variety of symbols to represent a particular landscape characteristic, a cultural temple, than did the girls. Finally, we hypothesize that the ‘hunter–gatherer’ social divisions of labor between men and women in the village’s early historical social structure are consequentially related to gender discrepancies in cognitive mapping symbolic representation skills, in non-English-speaking children. (shrink)

Sound symbolism refers to non-arbitrary associations between word forms and meaning, such as those observed for some properties of sounds and size or shape. Recent evidence suggests that these connections extend to emotional concepts. Here we investigated two types of non-arbitrary relationships. Study 1 examined whether iconicity scores (i.e. resemblance-based mapping between aspects of a word’s form and its meaning) for words can be predicted from ratings in the affective dimensions of valence and arousal and/or the discrete emotions of (...) happiness, anger, fear, disgust and sadness. Words denoting negative concepts were more likely to have more iconic word forms. Study 2 explored whether statistical regularities in single phonemes (i.e. systematicity) predicted ratings in affective dimensions and/or discrete emotions. Voiceless (/p/, /t/) and voiced plosives (/b/, /d/, /g/) were related to high arousing words, whereas high arousing negative words tended to include fricatives (/s/, /z/). Hissing consonants were also more likely to occur in words denoting all negative discrete emotions. Additionally, words conveying certain discrete emotions included specific phonemes. Overall, our data suggest that emotional features might explain variations in iconicity and provide new insight about phonemic patterns showing sound symbolic associations with the affective properties of words. (shrink)

This paper examines whether a classical model could be translated into a PDP network using a standard connectionist training technique called extra output learning. In Study 1, standard machine learning techniques were used to create a decision tree that could be used to classify 8124 different mushrooms as being edible or poisonous on the basis of 21 different Features (Schlimmer, 1987). In Study 2, extra output learning was used to insert this decision tree into a PDP network being trained on (...) the identical problem. An interpretation of the trained network revealed a perfect mapping from its internal structure to the decision tree, representing a precise translation of the classical theory to the connectionist model. In Study 3, a second network was trained on the mushroom problem without using extra output learning. An interpretation of this second network revealed a different algorithm for solving the mushroom problem, demonstrating that the Study 2 network was indeed a proper theory translation. (shrink)

Countless conceptual metaphors related to human experience have been identified and discussed in the literature. In most conceptual metaphors, a concrete, experiential sou...

What is the relationship between place and cultural production? How do we account for the interaction between the domain of spatiality and that of artistic expression? In particular, how might we conceptualize the connections between space and literature? Here, I attempt to map the principal ways in which the central thematic issues we associate with literary expression are related to questions about space and place. By elucidating these matters, I hope to arrive at a rationale for an approach to setting (...) out the links between symbolic expression and spatiality. Literature is envisaged here as one specific type of cultural expression, and the suggestion is that much of what applies to literature can be taken as... (shrink)

The ability to combine words into novel sentences has been used to argue that humans have symbolic language production abilities. Critiques of connectionist models of language often center on the inability of these models to generalize symbolically (Fodor & Pylyshyn, 1988; Marcus, 1998). To address these issues, a connectionist model of sentence production was developed. The model had variables (role‐concept bindings) that were inspired by spatial representations (Landau & Jackendoff, 1993). In order to take advantage of these variables, a novel (...) dual‐pathway architecture with event semantics is proposed and shown to be better at symbolic generalization than several variants. This architecture has one pathway for mapping message content to words and a separate pathway that enforces sequencing constraints. Analysis of the model's hidden units demonstrated that the model learned different types of information in each pathway, and that the model's compositional behavior arose from the combination of these two pathways. The model's ability to balance symbolic and statistical behavior in syntax acquisition and to model aphasic double dissociations provided independent support for the dual‐pathway architecture. (shrink)

It is widely agreed that irrelevant numerical values are automatically activated. However, automatic and intentional activations may give rise to different numerical representations. We examined processing of symbolic and non-symbolic representations asking whether they differ in automatic and intentional processing. Participants were presented with two-dimensional displays containing repetitions of a digit and were asked to report, in different blocks, whether the digit or numerosity was smaller or larger than 5. Incongruent trials differed either in laterality between the relevant and (...) irrelevant dimensions or in numerical distance between dimensions. Congruency affected performance regardless of symbolic or non-symbolic presentation. For incongruent trials, laterality affected performance, again regardless of presentation. This implies that automaticity does not mean similar processing of relevant and irrelevant dimensions. Specifically, the relevant dimension is processed elaborately whereas the irrelevant dimension is processed crudely. (shrink)

Nicolas Abraham (1919–75) rethinks the symbol as the very fabric of being. The author examines how this notion challenges the limitations of Husserl’s phenomenology and its reliance on a transcendental ego that can apprehend hyletic data in its purity. For Abraham, the symbol is worldly and resonates with its emergence from intersubjective foundations to constitute subjectivity impurely as a Dyad. It is born from trauma, a cut that differentiates Ego from Other but also generates anxiety (and Time) to (...) keep its operation unfinished. This projection into the future implicates intention (agency) that builds upon origins that are opaque and immemorial. Abraham denotes this origin as the Arche, conceiving of it as a primary symbolism that renders the traumatic cut but without temporal or structural guarantee. It is a beginning that is beyond the phenomenal, requiring exploration through a radical transphenomenological perspective derived from psychoanalysis. Here, trauma is both the debilitation of the symbol and (potentially) the impetus for its transformation, (re)iterating structures of subjectivity that have greater complexity and degrees of freedom. This is, of course, as circumstances allow. Abraham’s construction of symbolic subjectivity is stratified as the symbol negotiates levels of existence from basic Ego-integration to sophisticated social organizations, none of which can individually capture its operation. The journey to a fulfilling existence is riddled with obstructions that could be physiological, relational, cultural or otherwise. The author highlights this multivalence as a complexity that must be carefully navigated – especially when there is suffering – by attending to the symbol and its resonances where creative transformations cannot be mapped in advance. (shrink)

The mind maps symbols and the extra-symbolic relationships amongst them to specific meanings. When symbols of various levels are placed in a hierarchical ordering, one may look at such ordered classes as distinct worlds where one class represents objects and the other represents the objects’ corresponding meanings. However, such an explanation can only be partial because the number of potential levels in such an ordering is infinite and, therefore, it engenders problems of recursion and infinite regress. There are also logical (...) problems in the form of paradoxes that emanate from the consideration of sets of sets. Given that most prior studies only consider symbols that are classical objects in associative relationships, we argue that there is a need to also consider objects with shifting boundaries and quantum objects. We believe that objects belonging to each of these three classes—that is classical objects, objects with shifting boundaries, and quantum objects—play a role in the workings of the mind. (shrink)

Although language has long been regarded as a primarily arbitrary system, sound symbolism, or non-arbitrary correspondences between the sound of a word and its meaning, also exists in natural language. Previous research suggests that listeners are sensitive to sound symbolism. However, little is known about the specificity of these mappings. This study investigated whether sound symbolic properties correspond to specific meanings, or whether these properties generalize across semantic dimensions. In three experiments, native English-speaking adults heard sound symbolic foreign words for (...) dimensional adjective pairs and for each foreign word, selected a translation among English antonyms that either matched or mismatched with the correct meaning dimension. Listeners agreed more reliably on the English translation for matched relative to mismatched dimensions, though reliable cross-dimensional mappings did occur. These findings suggest that although sound symbolic properties generalize to meanings that may share overlapping semantic features, sound symbolic mappings offer semantic specificity. (shrink)

Exponential mappings into an imaginary space or number field for the axioms of a theory, which are in the form of propositional constants and variables, make possible: (a) an understanding of the meaning and differences between the Lorentz transformation constants, such that their product is still equal to one, but the axioms at each end of the transformations are logically inverse and separately consistent; (b) an interpretation of the psi function phase factor which is part of the axiomE=hf; (c) the (...) unification of the quantum-mechanical psi function and the electromagnetic wave function. Thus, those statements whose mechanisms are unknown (the axioms of the theory) are to be assigned the axiom propositional number symbol ϑ and are to be associated with the complex probability eiϑ, which is a uniform factor of the energy equations expressing the physical state. Such probabilistic axiom functions can be associated with both the special theory of relativity and the quantum-electromagnetic theory. (shrink)

In this paper we show that it is consistent with ZFC that for any set of reals of cardinality the continuum, there is a continuous map from that set onto the closed unit interval. In fact, this holds in the iterated perfect set model. We also show that in this model every set of reals which is always of first category has cardinality less than or equal to ω 1.

Spatial mental representations can be derived from linguistic and non‐linguistic sources of information. This study tested whether these representations could be formed from statistical linguistic frequencies of city names, and to what extent participants differed in their performance when they estimated spatial locations from language or maps. In a computational linguistic study, we demonstrated that co‐occurrences of cities in Tolkien’s Lord of the Rings trilogy and The Hobbit predicted the authentic longitude and latitude of those cities in Middle Earth. In (...) a human study, we showed that human spatial estimates of the location of cities were very similar regardless of whether participants read Tolkien’s texts or memorized a map of Middle Earth. However, text‐based location estimates obtained from statistical linguistic frequencies better predicted the human text‐based estimates than the human map‐based estimates. These findings suggest that language encodes spatial structure of cities, and that human cognitive map representations can come from implicit statistical linguistic patterns, from explicit non‐linguistic perceptual information, or from both. (shrink)

This study examined the metaphorical expressions of death in Chinese quadrisyllabic idioms. Specifically, the research investigated the cultural connotations and implications conveyed through death-related idioms by analyzing metaphorical examples. Adopting the ‘Great Chain of Being’ framework (Lakoff and Turner, 1989), a total of 579 death-related idioms with metaphorical and euphemistic meanings were classified and examined. These idioms were further categorized based on the gender of the deceased as well as the initial metaphors and expressions employed. Our research findings highlight cultural (...) parallels evident in death-related idiomatic expressions. Both English and Chinese languages employ metaphors like “DEATH IS SLEEP” and “DEATH IS DEPARTURE.” These pervasive metaphors can be attributed to shared aspects of the human experience. Meanwhile, our findings also revealed a strong intertwining of metaphors and culture within death-related idioms, such that the metaphor “TO DIE IS TO FALL” is frequently used in Mandarin but is not as common in English. The metaphorical mapping mechanisms employed in these idioms reflect the Chinese way of projecting thoughts onto objects and highlight the unique features of the Chinese language. Chinese culture places great emphasis on social classes, relationships, and hierarchical structures. These notions are not only reflected in social etiquette and how one handles interactions, but are also evident in euphemisms and idioms associated with death. (shrink)

A wide range of systems appear to perform computation: what common features do they share? I consider three examples, a digital computer, a neural network and an analogue route finding system based on soap-bubbles. The common feature of these systems is that they have autonomous dynamics — their states will change over time without additional external influence. We can take advantage of these dynamics if we understand them well enough to map a problem we want to solve onto them. Programming (...) consists of arranging the starting state of a system so that the effects of the system''s dynamics on some of its variables corresponds to the effects of the equations which describe the problem to be solved on their variables. The measured dynamics of a system, and hence the computation it may be performing, depend on the variables of the system we choose to attend to. Although we cannot determine which are the appropriate variables to measure in a system whose computation basis is unknown to us I go on to discuss how grammatical classifications of computational tasks and symbolic machine reconstruction techniques may allow us to rule out some measurements of a system from contributing to computation of particular tasks. Finally I suggest that these arguments and techniques imply that symbolic descriptions of the computation underlying cognition should be stochastic and that symbols in these descriptions may not be atomic but may have contents in alternative descriptions. (shrink)

We argue that analogical reasoning, particularly Gentner's structure-mapping theory, provides an integrative theoretical framework through which we can better understand the development of symbol use. Analogical reasoning can contribute both to the understanding of others’ intentions and the establishment of correspondences between symbols and their referents, two crucial components of symbolic understanding. We review relevant research on the development of symbolic representations, intentionality, comparison, and similarity, and demonstrate how structure-mapping theory can shed light on several ostensibly disparate (...) findings in the literature. Focusing on visual symbols, we argue that analogy underlies and supports the understanding of both intention and correspondence, which may enter into a reciprocal bootstrapping process that leads children to gain the prodigious human capacity of symbol use. (shrink)

: Fodor argues that perceptual representations are a subset of iconic representations, which are distinguished from symbolic/discursive representations. Iconic representations are nonconceptual and they do not support the abilities afforded by concepts. Iconic representations, for example, cannot support object individuation. If someone thinks that perception or some of its parts has imagistic NCC, they face the following dilemma. Either they will have to accept that this NCC does not allow for object individuation, but it represents instead conglomerations of properties and (...) at some stage of visual processing it must interface with cognition and its conceptual capacities for the visual objects to be individuated. Or, they will have to hold that the imagistic, NCC of perception, allows for object individuation. I opt for the second thesis because I think there is strong empirical evidence that objects are individuated during early vision. I also think that early vision individuates objects by means of, what I had previously called nonconceptual perceptual demonstrative reference. I argue, first, why Fodor’s view that iconic NCC does not enable object individuation is false. I also argue, contra Fodor, that early vision allows the perception of the cardinality of sets of objects. Keywords : Early Vision; Analog Representations; Object Individuation; Arithmetic Cognition; Cardinality of Sets L'individuazione di oggetti mediante il contenuto iconico: come è rappresentata la numerosità nella rappresentazione iconica? Riassunto: Per Fodor le rappresentazioni percettive sono un sottoinsieme delle rappresentazioni iconiche, distinte dalle rappresentazioni simbolico/discorsive. Le rappresentazioni iconiche sono nonconcettuali e non supportano le abilità richieste dai concetti. Le rappresentazioni iconiche, per esempio, non supportano l'individuazione di oggetti. Se si pensa che la percezione o qualche sua parte abbia un contenuto nonconcettuale come immagine, si ci si imbatte nel seguente dilemma. O si accetta che il NCC non permetta di individuare oggetti, ma che rappresenti conglomerati di proprietà e che si interfacci con la cognizione e le sue capacità concettuali, per individuare gli oggetti visivi. O si dice che l'immagine, il NCC, della percezione consenta di individuare oggetti. Propendo per la seconda tesi, poiché penso che ci sia solida evidenza empirica per cui gli oggetti vengono individuati durante le prime fasi della visione, in cui l’individuazione avviene mediante quanto definito come riferimento dimostrativo del contenuto nonconcettuale percettivo. Chiarirò le ragioni per cui la concezione di Fodor, per cui il contenuto nonconcettuale iconico non supporta l’individuazione di oggetti, è falsa. Inoltre, contra Fodor, sosterrò che le prime fasi della visione permettono la percezione della cardinalità di insiemi di oggetti. Parole chiave: Prime fasi della visione; Rappresentazioni analogiche; Individuazione di oggetti; Cognizione aritmetica; Cardinalità degli insiemi. (shrink)

The ancient dualism of a sensible and an intelligible world important in Neoplatonic and medieval philosophy, down to Descartes and Kant, would seem to be supplanted today by a scientific view of mind-in-nature. Here, we revive the old dualism in a modified form, and describe mind as a symbolic language, founded in linguistic recursive computation according to the Church-Turing thesis, constituting a world L that serves the human organism as a map of the Universe U. This methodological distinction of L (...) vs. U helps to understand how and why structures of phenomena come to be opposed to their nature in human thought, a central topic in Heideggerian philosophy. U is uncountable according to Georg Cantor’s set theory but Language L, based on the recursive function system, is countable, and anchored in a Gray Area within U of observable phenomena, typically symbols, prelinguistic structures, genetic-historical records of their origins. Symbols, the phenomena most familiar to mathematicians, are capable of being addressed in L-processing. The Gray Area is the human Environment E, where we can live comfortably, that we manipulate to create our niche within hostile U, with L offering overall competence of the species to survive. The human being is seen in the light of his or her linguistic recursively computational mind. Nature U, by contrast, is the unfathomable abyss of being, infinite labyrinth of darkness, impenetrable and hostile to man. The U-man, biological organism, is a stranger in L-man, the mind-controlled rational person, as expounded by Saint Paul. Noumena can now be seen to reside in L, and are not fully supported by phenomena. Kant’s noumenal cause is the mental L-image of only partly phenomenal causation. Mathematics occurs naturally in pre-linguistic phenomena, including natural laws, which give rise to pure mathematical structures in the world of L. Mathematical foundation within philosophy is reversed to where natural mathematics in the Gray Area of pre-linguistic phenomena can be seen to be a prerequisite for intellectual discourse. Lesser, nonverbal versions of L based on images are shared with animals. (shrink)

Cognitive theories based on a fixed feature set suffer from frame and symbol grounding problems. Flexible features and other empirically acquired constraints (e.g., analog-to-analog mappings) provide a framework for letting extrinsic relations influence symbol manipulation. By offering a biologically plausible basis for feature learning, nonorthogonal multiresolution analysis and dimensionality reduction, informed by functional constraints, may contribute to a solution to the symbol grounding problem.

This essay considers Moscow’s simultaneously peripheral and central position on the global fashion map. It is predicated on a study of imaginary Russian geographies presented in Vogue and other fashion media, advertisements and promotional activities by important fashion brands, as well as the promotional texts and visuals of several new Russian fashion designers. While these different players all contribute to shaping the imagery of Russian fashion today, their agendas and aesthetics differ. This essay identifies three main approaches within the field (...) of the symbolic production of Russian fashion. Western fashion designers and fashion media mainly rely on Russian imperial sartorial heritage in their orientalizing approach to Russian fashion. Secondly, Russian Vogue perpetuates Moscow’s peripheral international fashion position either by passively transmitting derivative Western representations of Russianness, or by reconstructing its own high-fashion versions of traditional Russian decorative style. Finally, several young Russian fashion designers deconstruct both traditional Russian and socialist iconography, in a fundamentally new development for the country’s fashion scene. (shrink)

This article proposes a neurobiologically motivated theory of meaning as internal representation that holds on to the principle of compositionality, but negates the principle of semantic constituency. The approach builds on neurobiological findings regarding topologically structured cortical feature maps and the mechanism of object-related binding by neuronal synchronization. It incorporates the Gestalt principles of psychology and is implemented by recurrent neural networks. The semantics to be developed is structurally analogous to some variant of model-theoretical semantics. The semantics to be developed (...) is a neuro-emulative model-theoretical semantics of a first-order language. The model-theoretical semantics is merely denotational and does not imply anything about the structures of the mind or the underlying neural mechanisms that enable producing and comprehend meaningful expressions. The relation between a mental representation and its content is some form of causal-informational covariation. (shrink)

Analogy and similarity are central phenomena in human cognition, involved in processes ranging from visual perception to conceptual change. To capture this centrality requires that a model of comparison must be able to integrate with other processes and handle the size and complexity of the representations required by the tasks being modeled. This paper describes extensions to Structure-Mapping Engine since its inception in 1986 that have increased its scope of operation. We first review the basic SME algorithm, describe psychological (...) evidence for SME as a process model, and summarize its role in simulating similarity-based retrieval and generalization. Then we describe five techniques now incorporated into the SME that have enabled it to tackle large-scale modeling tasks: Greedy merging rapidly constructs one or more best interpretations of a match in polynomial time: O); Incremental operation enables mappings to be extended as new information is retrieved or derived about the base or target, to model situations where information in a task is updated over time; Ubiquitous predicates model the varying degrees to which items may suggest alignment; Structural evaluation of analogical inferences models aspects of plausibility judgments; Match filters enable large-scale task models to communicate constraints to SME to influence the mapping process. We illustrate via examples from published studies how these enable it to capture a broader range of psychological phenomena than before. (shrink)

People often solve spatially presented cognitive problems more easily than their nonspatial counterparts. We explain this phenomenon by characterizing space as an inter-modality that provides common structure to different specific perceptual modalities. The usefulness of spatial structure for knowledge processing on different levels of granularity and for interaction between internal and external processes is described. Map representations are discussed as examples in which the usefulness of spatially organized symbols is particularly evident. External representations and processes can enhance internal representations and (...) processes effectively when the same structures and principles can be implicitly assumed. (shrink)

Marxism's collapse in the twentieth century profoundly altered the style and substance of Western European radical thought. To build a more robust form of democratic theory and action, prominent theorists moved to reject revolution, abandon class for more fragmented models of social action, and elevate the political over the social. Acknowledging the constructedness of society and politics, they chose the "symbolic" as a concept powerful enough to reinvent leftist thought outside a Marxist framework. Following Maurice Merleau-Ponty's Adventures of the Dialectic, (...) which reassessed philosophical Marxism at mid century, Warren Breckman critically revisits these thrilling experiments in the aftermath of Marxism. The post-Marxist idea of the symbolic is dynamic and complex, uncannily echoing the early German Romantics, who first advanced a modern conception of symbolism and the symbolic. Hegel and Marx denounced the Romantics for their otherworldly and nebulous posture, yet post-Marxist thinkers appreciated the rich potential of the ambiguities and paradoxes the Romantics first recognized. Mapping different ideas of the symbolic among contemporary thinkers, Breckman traces a fascinating reflection of Romantic themes and resonances, and he explores in depth the effort to reconcile a radical and democratic political agenda with a politics that does not privilege materialist understandings of the social. Engaging with the work of Claude Lévi-Strauss, Cornelius Castoriadis, Claude Lefort, Marcel Gauchet, Ernesto Laclau, Chantal Mouffe, and Slavoj Žižek, Breckman uniquely situates these important theorists within two hundred years of European thought and extends their profound relevance to today's political activism. (shrink)

Marxism's collapse in the twentieth century profoundly altered the style and substance of Western European radical thought. To build a more robust form of democratic theory and action, prominent theorists moved to reject revolution, abandon class for more fragmented models of social action, and elevate the political over the social. Acknowledging the constructedness of society and politics, they chose the "symbolic" as a concept powerful enough to reinvent leftist thought outside a Marxist framework. Following Maurice Merleau-Ponty's Adventures of the Dialectic, (...) which reassessed philosophical Marxism at mid century, Warren Breckman critically revisits these thrilling experiments in the aftermath of Marxism. The post-Marxist idea of the symbolic is dynamic and complex, uncannily echoing the early German Romantics, who first advanced a modern conception of symbolism and the symbolic. Hegel and Marx denounced the Romantics for their otherworldly and nebulous posture, yet post-Marxist thinkers appreciated the rich potential of the ambiguities and paradoxes the Romantics first recognized. Mapping different ideas of the symbolic among contemporary thinkers, Breckman traces a fascinating reflection of Romantic themes and resonances, and he explores in depth the effort to reconcile a radical and democratic political agenda with a politics that does not privilege materialist understandings of the social. Engaging with the work of Claude Lévi-Strauss, Cornelius Castoriadis, Claude Lefort, Marcel Gauchet, Ernesto Laclau, Chantal Mouffe, and Slavoj Žižek, Breckman uniquely situates these important theorists within two hundred years of European thought and extends their profound relevance to today's political activism. (shrink)