The Geometries of Visual Space

Mark Wagner

• Publisher: Psychology Press
• Year: 2012
• Print ISBN: 9780805852523
• eBook ISBN: 9781136871856
• Topic: Psychology
• Subtopic: Cognitive Psychology & Cognition
• Index: Psychology

When most people think of space, they think of physical space. However, visual space concerns space as consciously experienced, and it is studied through subjective measures, such as asking people to use numbers to estimate perceived distances, areas, angles, or volumes. This book explores the mismatch between perception and physical reality, and describes the many factors that influence the perception of space including the meaning assigned to geometric concepts like distance, the judgment methods used to report the experience, the presence or absence of cues to depth, and the orientation of a stimulus with respect to point of view. The main theme of the text is that no single geometry describes visual space, but that the geometry of visual space depends upon the stimulus conditions and mental shifts in the subjective meaning of size and distance.

In addition, The Geometries of Visual Space:
*contains philosophical, mathematical, and psychophysical background material;
*looks at synthetic approaches to space perception including work on hyperbolic, spherical, and Euclidean geometries;
*presents a meta-analysis of studies that ask observers to directly estimate size, distance, area, angle, and volume;
*looks at the size constancy literature in which observers are asked to adjust a comparison stimulus to match a variety of standards at different distances away;
*discusses research that takes a multi-dimensional approach toward studying visual space; and
*discusses how spatial experience is influenced by memory.

While this book is primarily intended for scholars in perception, mathematical psychology, and psychophysics, it will also be accessible to a wider audience since it is written at a readable level. It will make a good graduate-level textbook on space perception.

1

Introduction Contrasting Visual, Experiential, and Physical Space

Time and space are the two pure forms of all sensible perception, and as such they make a priori synthetic propositions possible.
Immanuel Kant, Critique of Pure Reason

Contrasting Conceptions of Space

This book will investigate the properties of our visual perceptions of space. The concept of space has been an object of speculation and dispute throughout the history of philosophy and science. Great philosophers and scientists—Immanuel Kant, Thomas Reid, Henri Poincaré, Issac Newton, and Albert Einstein (to name a few)—have considered space (together with time) to be one of the cornerstones on which existence is based and from which philosophy and science arise.

At the outset, two terms need to be defined and distinguished: physical space and visual space. While it is tempting to distinguish between the two by saying that the latter reflects conscious experience and the former does not, I believe one must resist this temptation. Both concepts reflect aspects of our experience of the world. But the attitude we take toward that experience differs between physical and visual space. Perhaps, I risk offending some readers by reviving the shadows of Wundt and Titchener, but in perception at least, Titchener was accurate in observing:

All human knowledge is derived from human experience; there is no other source. But human experience, as we have seen may be considered from different points of view…. First, we will regard experience as altogether independent of any particular person; we will assume that it goes on whether or not anyone is there to have it. Secondly, we will regard experience as altogether dependent on the particular person; we will assume that it goes on only when someone is there to share it. (Titchener, 1900/1909, p. 6)

However, conscious experience can be a rather slippery conceptual fish to grapple with. A scientist can only theorize based on solid data, and consciousness does not itself make marks on paper nor does it directly leave other physical traces that can be studied at leisure.

Harvey Carr once noted, “Consciousness is an abstraction that has no more independent existence than the grin of a Cheshire cat” (Carr, 1925). To open our experience to scientific investigation, we must rely on objectively observable behaviors and verbal reports that attempt to capture some aspect of experience, and we must resort to operational definitions of our concepts in order to render them concrete enough to use.

With this in mind, let me define my terms. By physical space I mean the space revealed to us by measuring devices such as rulers and protractors. Physical space is objectively defined; that is, the properties of physical space are largely observer independent. By visual space, I mean the space revealed by the psychophysical judgments of an observer. Visual space is not objectively defined; that is, the properties of visual space may depend critically on certain aspects of the observer, such as location in physical space, experimental conditions, and the mindset of the observer.

Defining visual space this way sidesteps the central issue: do the judgments people give accurately reflect their subjective experience of the world? Are the introspective reports that people generate a fair reflection of what is really witnessed internally? No doubt I would be wisest to simply drop the issue; however, I am too much of a philosopher to pass on without venturing an opinion.

Let me boldly state my own equivocal belief. While I believe that observers do attempt to base their judgments on their subjective experience of the world and I believe they really do try to be accurate, it is impossible to say how well they accomplish their goal. It is impossible to independently verify what is really in the subjective experience of an observer. The closest proxies we have are the judgments themselves.

Of course, if we did not believe that the numbers generated in psychophysical experiments reflected something of a person’s internal experience, we would quickly lose interest in the subject. Why would one really care about mere number generating responses? A true behaviorist should find perception boring.

Geometry and space. A variety of geometries have been employed to describe physical space at different levels of scale. When the distances under consideration are large, Einstein (1922) pointed out that a hyperbolic geometry might best describe physical space. When the slightly less grandiose distances of the earth’s surface are considered, a spherical (or elliptical) geometry makes sailing or flying around the world quicker and more efficient. Yet, if we confine ourselves to that range of distances which humans commonly experience; that is, if we confine ourselves to the ecological level of analysis mentioned by Gibson (1979); then any curvature in the earth’s surface or in the fabric of space itself is small enough to be ignored. The world is Euclidean. When distances are measured by a ruler, the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two legs to a high degree of approximation—just as Euclidean geometry would predict. When angles are measured by a protractor, the sum of the angles of any triangle is always very, very, very close to 180°—just as Euclidean geometry would predict.

The same definite conclusions cannot be made regarding visual space. People are capable of thinking about geometric concepts in different ways. By a simple mental shift, we can think of the distance from home to work as the crow flies, as the length of the path to get there, as the time it takes to drive, or as a segment of the “great circle” that intersects the two points. We can think of distance as the physicist sees it or take the artist’s perspective and see distance as the amount of canvas lying between two objects in a painting. One time we can use category estimation to judge distance and try to keep differences between categories subjectively identical while another time we use magnitude estimation and try to reflect the ratio of the subjective sizes of targets; and emphasizing these different mathematical aspects of the situation leads to very different psychophysical functions. Which of the geometries of visual space that result from these different perspectives is correct? I believe it is best to simply admit that no single view is correct, but that they all are. All may be valid descriptions of our varying subjective experience.

In addition, our experience is influenced by the situation we find ourselves in. Trying to judge the distance to an on-coming car is more difficult at night than it is during the day. Things that are far away can seem different than when they are brought close to us, and the angle from which we regard an object can make a difference to our perceptions of it. The world can seem large in the mind of a child, but the adult who returns to the old neighborhood is struck by how small and underwhelming things seem.

As we will see later, many have attempted to specify the geometry of visual space, but in my view that enterprise is hopeless from the outset. There is no single geometry that describes visual space, but there are many. The geometries of visual space vary with experience, with mental set, with conditions, and with time.

The purpose of this book is to determine how the geometry of visual space changes along with conditions. In addition, as part of that, this book will look at the changing relationship that exists between physical space and our visual perceptions of it.

Memory and space. The foregoing hints that physical and visual spaces are not the only ones of interest to the psychologist. What of memorial space, space as we remember it based on a past viewing of an object or setting? Even if one believed that space as it is directly perceived is both accurate and Euclidean as a Gibsonian would suggest, a psychologist would have good reason to suppose that the process of memory would distort our judgments into a very non-Euclidean form. Memories are incomplete and reconstructed.

Cognitive maps are another step away from direct perception. Cognitive maps refer to our mental representations of the layout of our surrounding environment. Cognitive maps generally concern large-scale environments that are too big to ever be seen at one time (except perhaps from an airplane or a space ship); so, cognitive maps are constructed across time based on our unfolding experience. As we will see later, cognitive maps are riddled with holes (that represent unexperienced territories), distortions, discontinuities, and non-spatial associations. A complete characterization of cognitive space is not only non-Euclidean; it is probably non-Riemannian. In fact, there may be no simple mathematical system that could ever fully characterize the richly chaotic nature of our cognitive maps. Cognitive maps may consist of a patchwork of loosely connected parts.

From a psychological standpoint, memorial space and cognitive maps certainly deserve our attention, and this book will describe something of their nature. Once more, the family of geometries that describe human experiences expands. Who could think there might be only one?

Experiential space. Of course, one need not stop here. A more general conception than visual space is that of experiential space. By experiential space I refer to our experience of space of any kind. By its very nature, the term visual space excludes spatial perceptions based on the other senses. Yet, clearly we do perceive space in extra-visual ways. Not only does it make sense to speak of visual space, but one may also meaningfully speak of auditory space, haptic space, gustatory space, kinesthetic space, proprioceptive space, and olfactory space. This book will largely confine itself to vision because the vast majority of research studies on spatial perception concern visual stimuli, but I will have a few words to say about these other spaces in various places in this book.

Why Is This Problem Important?

A noble intellectual pedigree. The problem of space perception is one with a long and prestigious pedigree. According to Wade (1996), ancient Greek philosophers including Aristotle and Euclid recognized that spatial perception did not always correspond to physical measures and that variables such as binocularity, aerial perspective, and distance to the stimulus can alter size estimates. Roman era thinkers including Galen, Lucretius, and Ptolemy noted that variables like linear perspective and the orientation of a stimulus can lead to breakdowns in size constancy. The great 11th Century Islamic philosopher, Ibn al-Haytham, spoke of the effects of stereopsis and familiar size on spatial perception. Leonardo da Vinci reiterated the importance of binocularity and aerial perspective on size perception.

Philosophers throughout the modern era often wrote about spatial experience as part of their systems of philosophy. Wade (1996) mentions Francis Bacon’s and René DesCartes’s interest in the problems of space perception. As will be discussed at length in Chapter 2, Berkeley, Hume, Kant, Reid, Poincaré, and Husserl all held well-developed views on the geometric character of our spatial experience.

Interest in the problem of space perception also played an integral role in the development of psychology as a discipline. Helmholtz (1868/1921) extensively wrote about space perception and empirically investigated the problem as part of his assault on Kantian philosophy. Weber’s studies of two-point limen in touch were largely motivated by his wish to understand how humans develop our sense of space. Other early founders of psychology, including Titchener and James, wrote extensive chapters (or even multiple chapters) on space perception in their foundational works on psychology. In fact, the longest single chapter in James’s two-volume The Principles of Psychology is dedicated to the subject. Wundt, whom some consider the founder of psychology, was so dedicated to studying the nature of space perception that James said of him: “Wundt has all his life devoted himself to the elaboration of space theory” (James, 1890, p. 276). (By the way, I tend to agree with Link (1994, 2002) that Fechner is a better candidate for the role of psychology’s founder than Wundt. While Wundt may have been better at self-promotion, psychology was alive and well before he ever came on the scene.)

Harvey Carr (1935), the great American Functionalist, wrote an entire book on space perception. In addition, when G. Stanley Hall was granted the first Ph.D. ever awarded in psychology in America, his dissertation was on (you guessed it) space perception (Boring, 1950).

In short, some of the greatest philosophers and psychologists in history focused considerable attention on the problems of space perception. The present book follows this rich tradition and reconceptualizes our spatial experience in the light of the massive body of empirical research performed in more recent years.

Space is foundational. These great minds devoted so much of their attention of spatial experience for a very good reason. Space is foundational. The universe itself may represent little more than the interplay of space, time, and energy. Modern physics seeks to explain gravity, black holes, and the expansion of the universe in terms of alterations in the fabric of space.

Psychologically, space is one of the fundamental building blocks of human experience. Without a conception of space, object perception and meaningful interaction with the world would be impossible. One literally could not live without some ability to sense the layout of the world. At times, one literally cannot live when this perception is in error at a critical time.

Kant (1781/1929) firmly believed that spatial experience served as the base out of which our phenomenal experience grows. In his words:

Space is a necessary a priori representation, which underlies all outer intuitions. We can never represent to ourselves the absence of space, though we can quite well think it as empty of objects. It must therefore be regarded as the condition of the possibility of appearances, and not as a determination dependent on them. (Kant, 1781/1929, p. 24)

Like Kant, I feel that spatial experience represents something particularly fundamental that deserves detailed study. Unlike Kant, I believe that explicating the nature of visual space is an empirical, rather than a logical, a priori issue. This book describes the nature of visual space as revealed by the research literature.

A paradigm for measuring mind. Fechner (1860) and Wundt (1874/1904) attempted to apply mathematical tools and the scientific method to the study of consciousness, and for a while all of psychology focused on the study of conscious mind. But as time passed, psychology became ever less interested in consciousness and ever more interested in behavior. Why did this happen? Some believe early Structural Psychology died due to its methodological defects. Carr (1925), who did not wholly reject the introspective method of the Structuralists, pointed out the defects of introspection. He felt that introspection was too difficult to do to give much detailed information about consciousness, that introspective reports were not subject to independent verification, and that Structuralists tended to rely on trained observers whose observations were too easily influenced by their knowledge of the research hypotheses—what James (1890) referred to as the Psychologist’s Fallacy.

A more fatal line of attack on introspection came from Watson (1914, 1919, 1924, 1925). Watson felt that it was impossible to make any real progress with a science based on introspection and that the whole enterprise could be dismissed as irrelevant. “The psychology begun by Wundt has thus failed to become a science and has still more deplorably failed in contributing anything of a scientifically usable kind to human nature” (Watson, 1919, p. 3).

While I realize that modern psychology has lost much of its behaviorist character, Watson’s challenge is still one I take very seriously. Is it possible to take introspective reports and develop them into an organized, sophisticated, developing body of knowledge? If Watson is right, then it is not only difficult to study the mind, but mind becomes a mere wisp or vapor of no importance.

But, one can develop a sophisticated science based on introspective reports. And I believe no area of psychology is fitter to demonstrate this point than the spatial perception literature. Space perception can be seen as a paradigm of success in the study of mind. This book is an attempt to answer Watson’s charge.

More recently, a second serious charge was leveled against the whole enterprise of psychophysics. Lockhead (1992) accused psychophysicists of generating a sterile discipline that consists of a series of unidimensional investigations that fail to adequately grapple with the effects of context on judgments. I see the present book as a lengthy refutation of Lockhead’s charge. When taken together the spatial perception literature paints a rich, multidimensional picture that dynamically changes as a function of contextual variables.

Practical applications of visual space perception. James (1907/1964) felt that scientists could be divided into two groups based on their temperaments. The forgoing justifications might appeal to those with what James referred to as a “tender-minded make-up,” but might not convince those with a more “tough-minded make-up.” A final justification for the study of space perception might even satisfy readers of the hard-nosed persuasion. Space perception research can have many prac...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Dedication
• Contents
• Preface
• 1 Introduction: Contrasting Visual, Experiential, and Physical Space
• 2 Traditional Views of Geometry and Vision
• 3 Synthetic Approaches to Visual Space Perception
• 4 An Analytic Approach to Space and Vision
• 5 Effects of Context on Judgments of Distance, Area, Volume, and Angle
• 6 Factors Affecting Size Constancy
• 7 The Metrics of Visual Space: Multidimensional Approaches to Space Perception
• 8 Cognitive Maps, Memory, and Space Perception
• 9 The Geometries of Visual Space: Conclusion
• References
• Author Index
• Subject Index