Remarks on the Geometry of Visibles

I offer an explication of Reid's claim (discussed recently by Yaffe and others) that the geometry of the visual field is spherical geometry. I show that the sphere is the only surface whose geometry coincides, in a certain strong sense, with the geometry of visibles.     

Spatial behavior: the impact of global and local geometry

Humans and other animals use the global geometry of the surrounding environment in order to orient and determine which direction they are facing. Accordingly, the impact of environment geometry on spatial behavior is reflected in the paths of progression in the environment. When the perception of the global geometry is limited, such as in large or dark environments, the global geometry of the environment has to be constructed gradually as the accumulated geometry of locales. In the present study, we progressively altered the form of a dark square test arena by means of local alterations to its corners and walls, in order to differentiate the impact of the global arena geometry from that of the local arena geometry sectors on spatial behavior in rats. We found that as long as the local alterations did not distort the global square geometry of the dark environment, the rats’ behavior did not significantly change. In contrast, distortion of the square shape of the arena resulted in significant changes in the spatial distribution of the rats’ activity. Accordingly, we suggest that the perceived global geometry affects spatial behavior, overriding the impact of the local geometry.     

Early sex differences in weighting geometric cues

When geometric and non-geometric information are both available for specifying location, men have been shown to rely more heavily on geometry compared to women. To shed insight on the nature and developmental origins of this sex difference, we examined how 18- to 24-month-olds represented the geometry of a surrounding (rectangular) space when direct non-geometric information (i.e. a beacon) was also available for localizing a hidden object. Children were tested on a disorientation task with multiple phases. Across experiments, boys relied more heavily than girls on geometry to guide localization, as indicated by their errors during the initial phase of the task, and by their search choices following transformations that left only geometry available, or that, under limited conditions, created a conflict between beacon and geometry. Analyses of search times suggested that girls, like boys, had encoded geometry, and testing in a square space ruled out explanations concerned with motivational and methodological variables. Taken together, the findings provide evidence for an early sex difference in the weighting of geometry. This sex difference, we suggest, reflects subtle variation in how boys and girls approach the problem of combining multiple sources of location information.     

From Inexactness to Certainty: The Change in Hume's Conception of Geometry

Although Hume's analysis of geometry continues to serve as a reference point for many contemporary discussions in the philosophy of science, the fact that the first Enquiry presents a radical revision of Hume's conception of geometry in the Treatise has never been explained. The present essay closely examines Hume's early and late discussions of geometry and proposes a reconstruction of the reasons behind the change in his views on the subject. Hume's early conception of geometry as an inexact non-demonstrative science is argued to be a consequence of his attempt to discredit geometrical proofs of infinite divisibility of extension by anchoring the meaning of geometrical concepts in inherently inexact qualitative measurement procedures. This measurement-based attack on the exactness and certainty of geometry is analyzed and shown to be both self-refuting and inconsistent with the general epistemological framework of the Treatise. The revised conception of geometry as a demonstrative science in the first Enquiry is then interpreted as Hume's response to the failure of his earlier attempt to discredit geometrical proofs of infinite divisibility of extension. This revised version was published online in August 2006 with corrections to the Cover Date.     

Surface geometry influences the shape of illusory contours

Geometric and neural models of illusory-contour (IC) synthesis currently use only local contour geometry to derive the shape of ICs. Work on the visual representation of shape, by contrast, points to the importance of both contour and surface geometry. We investigated the influence of surface-based geometric factors on IC shape. The local geometry of inducing-contour pairs was equated in stereoscopic IC displays, and the shape of the enclosed surface was varied by manipulating sign of curvature, cross-axial shape width, and medial-axis geometry. IC shapes were measured using a parametric shape-adjustment task (Experiment 1) and a dot-adjustment task (Experiment 2). Both methods revealed large influences of surface geometry. ICs enclosing locally concave regions were perceived to be systematically more angular than those enclosing locally convex regions. Importantly, the influence of sign of curvature was modulated significantly by shape width and medial-axis geometry: IC shape difference between convex and concave inducers was greater for narrow shapes than wider ones, and greater for shapes with straight axis and symmetric contours (diamond versus bowtie), than those with curved axis and parallel contours (bent tubes). Even at the level of illusory "contours," there is a contribution of region-based geometry which is sensitive to nonlocal shape properties involving medial geometry and part decomposition. Models of IC synthesis must incorporate the role of nonlocal region-based geometric factors in a way that parallels their role in organizing visual shape representation more generally.     

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]     

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.     

STANDARDS OF EQUALITY AND HUME'S VIEW OF GEOMETRY

It has been argued that there is a genuine conflict between the views of geometry defended by Hume in the Treatise and in the Enquiry: while the former work attributes to geometry a different status from that of arithmetic and algebra, the latter attempts to restore its status as an exact and certain science. A closer reading of Hume shows that, in fact, there is no conflict between the two works with respect to geometry. The key to understanding Hume's view of geometry is the distinction he draws between two standards of equality in extension.     

History of Geometry and the Development of the Form of its Language

The aim of this paper is to introduce Wittgenstein’s concept of the form of a language into geometry and to show how it can be used to achieve a better understanding of the development of geometry, from Desargues, Lobachevsky and Beltrami to Cayley, Klein and Poincaré. Thus this essay can be seen as an attempt to rehabilitate the Picture Theory of Meaning, from the Tractatus. Its basic idea is to use Picture Theory to understand the pictures of geometry. I will try to show, that the historical evolution of geometry can be interpreted as the development of the form of its language. This confrontation of the Picture Theory with history of geometry sheds new light also on the ideas of Wittgenstein. This revised version was published online in June 2006 with corrections to the Cover Date.     

The theory‐ladenness of observations,the role of scientific instruments,and the Kantian a priori

During the last decades it has become widely accepted that scientific observations are ‘theory‐laden’. Scientists ‘see’ the world with their theories or theoretical presuppositions. In the present paper it is argued that they ‘see’ with their scientific instruments as well, as the uses of scientific instruments is an important characteristic of modern natural science. It is further argued that Euclidean geometry is intimately linked to technology, and hence that it plays a fundamental part in the construction and operation of scientific instruments. Finally, Euclidean geometry is compared to fractal geometry, and the question of its a priori status is raised. Although the position that Euclidean geometry is a priori in the original Kantian sense is untenable, the paper concludes that in some restricted sense Euclidean geometry may be said to be a priori.     

Learning about environmental geometry: an associative model

K. Cheng (1986) suggested that learning the geometry of enclosing surfaces takes place in a geometric module blind to other spatial information. Failures to find blocking or overshadowing of geometry learning by features near a goal seem consistent with this view. The authors present an operant model in which learning spatial features competes with geometry learning, as in the Rescorla-Wagner model. Relative total associative strength of cues at a location determines choice of that location and thus the frequencies of reward paired with each cue. The model shows how competitive learning of local features and geometry can appear to result in potentiation, blocking, or independence, depending on enclosure shape and kind of features. The model reproduces numerous findings from dry arenas and water mazes.     

The Simplest Axiom System for Hyperbolic Geometry Revisited,Again

Dependencies are identified in two recently proposed first-order axiom systems for plane hyperbolic geometry. Since the dependencies do not specifically concern hyperbolic geometry, our results yield two simpler axiom systems for absolute geometry.     

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.     

Internalization: a metaphor we can live without

Shepard has supposed that the mind is stocked with innate knowledge of the world and that this knowledge figures prominently in the way we see the world. According to him, this internal knowledge is the legacy of a process of internalization; a process of natural selection over the evolutionary history of the species. Shepard has developed his proposal most fully in his analysis of the relation between kinematic geometry and the shape of the motion path in apparent motion displays. We argue that Shepard has made a case for applying the principles of kinematic geometry to the perception of motion, but that he has not made the case for injecting these principles into the mind of the percipient. We offer a more modest interpretation of his important findings: that kinematic geometry may be a model of apparent motion. Inasmuch as our recommended interpretation does not lodge geometry in the mind of the percipient, the motivation of positing internalization, a process that moves kinematic geometry into the mind, is obviated. In our conclusion, we suggest that cognitive psychologists, in their embrace of internal mental universals and internalization may have been seduced by the siren call of metaphor.     

FORMALIZATION OF HILBERT'S GEOMETRY OF INCIDENCE AND PARALLELISM

Three things are presented: How Hilbert changed the original construction postulates of his geometry into existential axioms; In what sense he formalized geometry; How elementary geometry is formalized to present day's standards.     

A theory of phenomenal geometry and its applications

The geometry of perceived space (phenomenal geometry) is specified in terms of three basic factors: the perception of direction, the perception of distance or depth, and the perception of the observer's own position or motion. The apparent spatial locations of stimulus points resulting from these three factors thereupon determine the derived perceptions of size, orientation, shape, and motion. Phenomenal geometry is expected to apply to both veridical and illusory perceptions. It is applied here to explain a number of representative illusions, including the illusory rotation of an inverted mask (Gregory, 1970), a trapezoidal window (Ames, 1952), and any single or multiple point stimuli in which errors in one or more of the three basic factors are present. It is concluded from phenomenal geometry that the size-distance and motion-distance invariance hypotheses are special cases of the head motion paradigm, and that proposed explanations in terms of compensation, expectation, or logical processes often are unnecessary for predicting responses to single or multiple stimuli involving head or stimulus motion. Two hypotheses are identified in applying phenomenal geometry. It is assumed that the perceptual localization of stimulus points determines the same derived perceptions, regardless of the source of perceptual information supporting the localizations. This assumption of cue equivalence or cue substitution provides considerable parsimony to the geometry. Also, it is assumed that the perceptions specified by the geometry are internally consistent. Departures from this internal consistency, such as those which occur in the size-distance paradox, are considered to often reflect the intrusion of nonperceptual (cognitive) processes into the responses. Some theoretical implications of this analysis of phenomenal geometry are discussed.     

对物理学之几何化发展的哲学思考

物理学与几何学的关系密切:几何不仅是物理学家描述物理的数学手段,而且是启发他们物理思想的源泉。当物理学家重新考察几何学与物理学的关系时,物理学就会发生一次重大进步甚至革命。伽利略、牛顿的物理学革命将欧几里得几何引入,确立了描写我们世界的数学法则,奠定了经典物理     

不同认知风格学生的问题共性意识水平对解题迁移的影响

 本文通过实验,对初中生的平面几何成绩与其认知风格之间的关系、解题者对迁移题与源题之间共性关系的意识水平对解题迁移的影响进行了研究。结果表明:(1)初中生的平面几何成绩与认知风格有显著的正相关;(2)在平面几何解题过程中,当迁移题与源题之间具有结构共同性,且表面相似性和条件隐蔽性为中等程度时,解题者对先后问题之间共性关系的意识水平对解题迁移有显著影响。