Linear Measurement Models-Axiomatizations and Axiomatizability

This paper examines, in the scope of representational measurement theory, different axiomatizations and axiomatizability of linear and bilinear representations of ordinal data contexts in real vector spaces. The representation theorems proved in this paper are modifications and generalizations of Scott's characterization of finite linear measurement models. The advantage of these representation theorems is that they use only finitely many axioms, the number of which depends on the size of the given ordinal data context. Concerning the axiomatizability, it is proved by model-theoretic methods that finite linear measurement models cannot be axiomatized by a finite set of first order axioms. Copyright 2000 Academic Press.     

On Independent Axiomatizability of Quasi-Normal Modal Logics

Studia Logica - We give a negative solution to the problem, posed by A. Chagrov and M. Zakharyaschev, of whether every quasi-normal propositional modal logic can be axiomatized by an independent...     

Acquisition of locational information about reference points during locomotion with and without a concurrent task: Effects of number of reference points

Maintenance of orientation relative to reference points along a locomotion path and acquisition of locational information about these reference points were investigated by requiring subjects with and without a concurrent backwards counting task to walk in alleys and estimate direction and distance to the reference points when stopped. One or three reference points were presented on each trial in different groups of subjects. Acquisition of locational information was indicated by decreased latencies and variable errors with repeated exposure to the same locomotion path. Neither the larger number of reference points nor the concurrent task seemed to affect acquisition, but both lead to increased latencies indicating that the subjects under these conditions were unable to keep track of the locations of the reference points whilst walking. However, they may still have been able to infer these locations after the locomotion path had been traversed.     

Talking points: A modulating circle reduces listening effort without improving speech recognition

Psychonomic Bulletin & Review - Speech recognition is improved when the acoustic input is accompanied by visual cues provided by a talking face (Erber in Journal of Speech and Hearing Research,...     

The organization of room geometry and object layout geometry in human memory

Research with humans and with nonhuman species has suggested a special role of room geometry in spatial memory functioning. In two experiments, participants learned the configuration of a room with four corners, along with the configuration of four objects within the room, while standing in a fixed position at the room's periphery. The configurations were either rectangular (Experiment 1) or irregular (Experiment 2). Room geometry was not recalled better than object layout geometry, and memories for both configurations were orientation dependent. These results suggest that room geometry and object layout geometry are represented similarly in human memory, at least in situations that promote long-term learning of object locations. There were also some differences between corners and objects in orientation dependence, suggesting that the two sources of information are represented in similar but separate spatial reference systems. [corrected]     

Hans Reichenbach's relativity of geometry

Hans Reichenbach's 1928 thesis of the relativity of geometry has been misunderstood as the statement that the geometrical structure of space can be described in different languages. In this interpretation the thesis becomes an instance of trivial semantical conventionalism, as Grünbaum calls it. To understand Reichenbach correctly, we have to interpret it in the light of the linguistic turn, the transition from thought oriented philosophy to language oriented philosophy, which mainly took place in the first decades of our century. Reichenbach — as Poincaré before him — is undermining the prejudice of thought oriented philosophy, that two propositions have different factual content, if they are associated with different ideas in our mind. Thus Reichenbach prepared the change to language oriented philosophy, which he also accepted later.     

The geometry of factorial indeterminancy

TheN-dimensional geometry of a Spearman-Thurstone factor solution reveals two sources for the indeterminancy of factor scores: indeterminancy of total factor space and a rotational indeterminancy within a given total factor space. The analytical papers of Ledermann [4] and Guttman [2] on indeterminancy of factor scores are related to these findings and a simple vector model is developed to reveal the properties of rotational indeterminancy. The significance of factor-score indeterminancy is discussed in light of these findings.     

Core geometry in perspective

Research on animals, infants, children, and adults provides evidence that distinct cognitive systems underlie navigation and object recognition. Here we examine whether and how these systems interact when children interpret 2D edge‐based perspectival line drawings of scenes and objects. Such drawings serve as symbols early in development, and they preserve scene and object geometry from canonical points of view. Young children show limits when using geometry both in non‐symbolic tasks and in symbolic map tasks that present 3D contexts from unusual, unfamiliar points of view. When presented with the familiar viewpoints in perspectival line drawings, however, do children engage more integrated geometric representations? In three experiments, children successfully interpreted line drawings with respect to their depicted scene or object. Nevertheless, children recruited distinct processes when navigating based on the information in these drawings, and these processes depended on the context in which the drawings were presented. These results suggest that children are flexible but limited in using geometric information to form integrated representations of scenes and objects, even when interpreting spatial symbols that are highly familiar and faithful renditions of the visual world.     

From geometry to phenomenology

Richard Tieszen [Tieszen, R. (2005). Philosophy and Phenomenological Research, LXX(1), 153–173.] has argued that the group-theoretical approach to modern geometry can be seen as a realization of Edmund Husserl’s view of eidetic intuition. In support of Tieszen’s claim, the present article discusses Husserl’s approach to geometry in 1886–1902. Husserl’s first detailed discussion of the concept of group and invariants under transformations takes place in his notes on Hilbert’s Memoir Ueber die Grundlagen der Geometrie that Hilbert wrote during the winter 1901–1902. Husserl’s interest in the Memoir is a continuation of his long-standing concern about analytic geometry and in particular Riemann and Helmholtz’s approach to geometry. Husserl favored a non-metrical approach to geometry; thus the topological nature of Hilbert’s Memoir must have been intriguing to him. The task of phenomenology is to describe the givenness of this logos, hence Husserl needed to develop the notion of eidetic intuition. The author wishes to thank Academy of Finland for financial assistance that enabled her to work on this article.     

Cognitive reference points

Two methods were used to test the hypothesis that natural categories (such as colors, line orientations, and numbers) have reference point stimuli (such as focal colors, vertical and horizontal lines, and numbers that are multiples of 10) in relation to which other stimuli of the category are judged. In Experiment I, subjects placed pairs of stimuli into sentence frames consisting of linguistic “hedges” such as “— is essentially—.” Results were that the supposed reference stimuli were most often placed in the second (reference) slot. In Experiment II, the subject placed a stimulus in physical space to represent his feeling of the psychological distance of that stimulus from another spatially fixed stimulus. Results showed that, when supposed reference stimuli were fixed, other stimuli were placed closer to them than vice versa. The results have substantive implications for the understanding of internal structure of categories and methodological implications for the mapping of reference points, quantification of linguistic intuitions, and the assumption of symmetry in psychological distance judgments.     

Erratum to: The organization of room geometry and object layout geometry in human memory

Research with humans and with nonhuman species has suggested a special role of room geometry in spatial memory functioning. In two experiments, participants learned the configuration of a room with four corners, along with the configuration of four objects within the room, while standing in a fixed position at the room’s periphery. The configurations were either rectangular (Experiment 1) or irregular (Experiment 2). Room geometry was not recalled better than object layout geometry, and memories for both configurations were orientation dependent. These results suggest that room geometry and object layout geometry are represented similarly in human memory, at least in situations that promote long-term learning of object locations. There were also some differences between corners and objects in orientation dependence, suggesting that the two sources of information are represented in similar but separate spatial reference systems.     

Children's use of geometry for reorientation

Research on navigation has shown that humans and laboratory animals recover their sense of orientation primarily by detecting geometric properties of large-scale surface layouts (e.g. room shape), but the reasons for the primacy of layout geometry have not been clarified. In four experiments, we tested whether 4-year-old children reorient by the geometry of extended wall-like surfaces because such surfaces are large and perceived as stable, because they serve as barriers to vision or to locomotion, or because they form a single, connected geometric figure. Disoriented children successfully reoriented by the shape of an arena formed by surfaces that were short enough to see and step over. In contrast, children failed to reorient by the shape of an arena defined by large and stable columns or by connected lines on the floor. We conclude that preschool children's reorientation is not guided by the functional relevance of the immediate environmental properties, but rather by a specific sensitivity to the geometric properties of the extended three-dimensional surface layout.     

Features enhance the encoding of geometry

Successful navigation within an environment requires that the traveler establish the correct heading—a process referred to as orienting. Many studies have now shown that humans and non-human animals can use the geometric properties of an enclosure to orient. In the present study, two groups of Clark’s nutcrackers (Nucifraga columbiana) were trained, in a reference memory task, to find food hidden in one of four containers arranged to form a rectangular array. One group had unique objects placed next to each of the containers, whereas the second group had identical objects placed next to each of the containers. Here, I show for the first time that for the Clark’s nutcracker, the distinctive properties of these objects enhanced the encoding of the array’s geometry compared to the learning of geometric properties from an array of identical objects, which remained at chance after substantial amounts of training. Subsequent transformation tests showed that an object not associated with reward, but sharing the same geometric properties as the correct object, may have had inhibitory qualities. Furthermore, by systematically removing objects from the array, I show that although nutcrackers encoded the geometry of the array, they did not encode a complete featural representation of the objects within the array.     

Pigeons encode relative geometry.

Pigeons were trained to search for hidden food in a rectangular environment designed to eliminate any external cues. Following training, the authors administered unreinforced test trials in which the geometric properties of the apparatus were manipulated. During tests that preserved the relative geometry but altered the absolute geometry of the environment, the pigeons continued to choose the geometrically correct corners, indicating that they encoded the relative geometry of the enclosure. When tested in a square enclosure, which distorted both the absolute and relative geometry, the pigeons randomly chose among the 4 corners, indicating that their choices were not based on cues external to the apparatus. This study provides new insight into how metric properties of an environment are encoded by pigeons.     

Figure-ground perception and random geometry

Constructs of random geometry were applied to the problem of figure-ground perception. Random-dot images of black and white dots with various area fractions and tesselations (square and triangular lattices) were used as stimuli. The constructs of random geometry are correlation functions of n-th order and some functionals defined on them. The only parameter which is independent of the tesselation used is the first-order correlation which is the area fraction. It was first conjectured and then experimentally verified that figureground perception is not affected by the various tesselations used. Thus, figure-ground phenomena depend only on the area fraction of the white and black dots in the stimulus. There is a perceptual bias for white, that is, figure-ground reversal is easiest at 40 percent white-black area fraction. It was also experimentally shown that size-constancy prevails in figure-ground perception, but brightnes s-constancy does not.     

The brain a geometry engine

Summary According to Kant, spacetime is a form of the mind. If so, the brain must be a geometry engine. This idea is taken seriously, and consequently the implementation of space and time in terms of machines is considered. This enables one to conceive of spacetime as really embodied.     

Visual space as physical geometry

The geometrical incongruence between patterns in visual space and structures and patterns of activity in the visual cortex, long known to investigators, serves as a criterion for evaluating physical theories of visual space. The problem of determining the geometry of the visual world (visual geometry) is compared with that of determining the geometry of the physical world (physical geometry). Theories as to the possible physical locus of visual space, whether in the brain or elsewhere, are reviewed, analyzed, and criticized accordingly. It is concluded that on the basis of congruence alone it would be predicted that visual space is not to be found in the brain, even though it is seemingly linked to it causally, as experimental neurology and neurophysiology demonstrate. Alternative theories as to the nature of visual space are considered, but are also found to be inadequate in explaining visual space in terms acceptable to contemporary science.