Roger N. Shepard, Professor of Psychology at Stanford University, is a particularly appropriate recipient for a prize dedicated to the “Theoretical Foundations of Human Cognition”. Throughout his research career, Roger Shepard has been searching for theoretical foundations of a science of the mind. His work attempts to specify such foundations in the form of universal laws formulated in an explicit mathematical manner, derivable from first principles, and which apply to human and to animal behaviour under a variety of tasks and stimulus sets. His endeavour to combine mathematical and physical modelling with quantified psychological experimentation has resulted in extraordinary advances in psychology. His work has opened up new research avenues in domains as varied as visual and auditory perception, mental imagery, representation, learning, and generalization. It is no exaggeration to state that several generations of psychologists have been influenced by the imagination and rigor that he has brought to psychological investigation. Indeed, many of the research paradigms that he has invented, from multidimensional scaling to mental rotation or guided apparent motion, continue to play a central role in current psychological investigations.
Roger Shepard’s recent keynote address to the Psychonomic Society  provides insights into his intellectual trajectory. Initially fascinated by mechanics, geometry, and Newtonian and relativistic physics, disciplines in which he received undergraduate training at Stanford, Roger Shepard became increasingly interested in the possibility that the tools of physics and mathematics might provide insights into the organization of mental representations, including those that took place within his own mind when he mentally explored those abstract objects. Under the influence of Fred Attneave’s research on similarity judgments, and of memorable lectures by William Estes and George Miller, Roger Shepard decided to orient his career towards the investigation of the laws of psychology. The influence of his early geometrical and physical training is clearly perceptible in the elegant, formal character of his experimental and theoretical contributions.
Internal representational spaces. A highly influential early contribution was Roger Shepard’s development of the method of nonmetric multidimensional scaling [12, 13], which was later improved by Joseph Kruskal, his mathematician colleague from Bell Laboratories. This method provided a new means of recovering the internal structure of mental representations from qualitative measures of similarity. This was accomplished without making any assumptions about the absolute quantitative validity of the data, but solely based on the assumption of a reproducible ordering of the similarity judgements. As a data analysis tool, non-metric dimensional scaling has proven extremely useful in many areas of science, and is now part of all major statistical packages. In this respect, Shepard’s contribution follows a long list of psychologists whose research has lead to the creation of new mathematical and statistical tools useful to the scientific community at large, as initially exemplified by Francis Galton’s invention of correlation or by Charles Spearman’s work on rank correlation and factor analysis.
More than any other psychologist perhaps, Roger Shepard played a pivotal role in drawing attention to the highly regular structure of mental representations, which he depicted as multidimensional “representational spaces”. He successively applied his non-metric multidimensional scaling method to many dimensions, including color , the pitch of sounds , or even abstract dimensions such as number . In each case, a beautiful internal structure emerged: the color circle, the “double helix” of pitch with independent circles for octaves and fifths, the logarithmic number line. All of these inferred representations have received extensive subsequent validation using a broad array of methods.
Universal law of generalization. Derivation of the internal structure of mental representations by multidimensional scaling allowed Roger Shepard to progress on his lifelong quest for laws of generalization [9-11, 17], which he considered as “the most fundamental problem confronting learning theory” [20, page 5]. As he notes, any theory of learning must specify how what has been learned in one situation is generalized to another. Once inter-stimulus distances were measured on the inferred representational space, Shepard observed that the generalization data from many experiments on both human and animal became highly regular. In essentially all cases, the probability with which a response that had been learned to one stimulus was made to another stimulus followed an exponentially decaying function. This regularity was reported in Shepard’s celebrated Science article entitled “Towards a universal law of generalization for Psychological Science” . In it, Shepard showed how a elegant, general mathematical theory, based on simple Bayesian principles and the concept of “consequential region”, could account for the universal exponential law of generalization. The theory also explained why two metrics were observed for psychological space: for unitary stimuli with “integral” dimensions (such as the lightness and saturation of colors), measuring internal distances with the Euclidean metric provides the best predictor of generalization; for other, analyzable stimuli (such as shapes differing in size and orientation), the best-fitting metric was the “city-block” metric, also known as the L1-norm. Roger Shepard suggested that any organism that attempts to generalize according to optimal laws should be led by natural selection to adopt the exponentially decaying law with the stimulus-appropriate metric. The theory also provided an explanation for another universal law, the law of discriminative reaction time, which indicates that the time to discriminate between two stimuli falls off as the inverse of the inter-stimulus distance. Shepard’s elegant theorizing thus lead to the unification of many fundamental observations on generalization and discrimination tasks in a remarkably broad variety of stimuli, tasks, and species.
Mental transformations. Perhaps Shepard’s most universally renowned experimental contribution consists in his experiments with the mental rotation task, thoroughly reported in his classic 1982 book “Mental images and their transformations”, written with his collaborator Lynn Cooper . Considered some of the most elegant chronometric experiments in the history of psychology, these studies demonstrated that the comparison of two views of the same objects, displayed in different 3-dimensional orientations, involves a process of “mental rotation”: the object is successively represented internally at successive positions which progressively bring one view in alignment with the other. Thus, the response time is a highly regular, linear function of the angle of internal rotation. It might be thought that mental rotation is a mere metaphor, but with Lynn Cooper, Roger Shepard demonstrated its “psychological reality”, for instance by demonstrating that the presentation of probe stimuli at intermediate orientations receive an especially fast response if presented at precisely the time when the theory predicts that this intermediate orientation should be internally represented.
Mental rotation has become a standard tool of psychology, and is now being applied in a variety of domains, from the assessment of brain-lesioned patients and airplane pilots to the investigation of the neural coding of movements and their transformations [4, 8]. Shepard himself extended his work in several directions. He showed that the phenomenon of apparent motion, which is perceived when two shapes are successively flashed in different orientations, exhibits lawful relations of display duration and trajectory length analogous to those observed under mental rotation conditions [e.g. 3]. He also provided a theoretical account of these laws. In both cases, the object path could be predicted by an analysis of the geodesic paths in the six-dimensional manifold jointly determined by the Euclidean group of three-dimensional space and the symmetry group of each object. Shepard also demonstrated that the path of apparent motion could be distorted by the presentation of a curved grey cue , again yielding highly regular laws relating motion and path length.By demonstrating that mental images could be empirically measured, transformed and controlled with unexpected precision, Shepard’s mental rotation paradigm played a key role in the great mental imagery debate. The similarities between perception and visual imagery were also demonstrated in other tasks such as a figure-ground search task that Shepard studied with his colleague Podgorny [6, 7].
Musical cognition. Roger Shepard’s interest in the internal structure of representations also led him to invent new perceptual illusions. In particular, Shepard’s highly imaginative research in the domain of musical cognition led to the invention of the Shepard scale. This is a sequence of sounds (now known as “Shepard tones”) which are each composed of multiple tones in octave relations, with fading amplitudes at each end of the frequency scale. Listening to the Shepard scale gives the illusion of an ever-ascending pitch . This illusion is analogous to Penrose’s illusion of ever-ascending steps, made famous by M.C. Escher’s wood engraving Ascending and Descending. The Shepard tones have been subject to much further experimental work such as the “tritone paradox” explored by UCSD psychologist Diana Deutsch . With Carol Krumhansl , Shepard further explored the universal laws of musical perception, again observing that distance on the internal representation of pitch could account for experimental results on the perception of tonal hierarchies. His work provided elaborate tools with which to study issues of universality and cultural differences in music perception [e.g. 1].
Besides those musical creations, Roger Shepard’s vivid imagination led him to generate playful, yet insightful visual illusions. A collection of his visual inventions, which he drew himself with great artistic talent, were published in his book MindSights .
Second-order isomorphism and internalization of physical laws. In recent syntheses of his work, Shepard has proposed an evolutionary psychology argument for why internal representations and their transformations are so regularly organized and often faithfully reflect the structure of physical laws [16, 19, 20]. He proposes that mental representations have evolved over millions of years as adaptations to universal physical principles (such as the kinematic laws governing object motion, those underlying light reflection and diffusion, etc). As a result, mental representations have become highly structured and attuned to physical laws –in Shepard’s terms they are “second-order isomorphic”, which means that the relations between physical events in the environment are preserved in the relations between their internal mental representations. According to Shepard, this mental internalization process explains why physicists such as Galileo, Newton or Einstein, were able to rely on thought experiments in order to derive plausible physical laws – thought processes are sufficiently isomorphic to physical processes that the properties of the latter can be inferred, in part, by mere introspection on the former. For Shepard, the mental regularities imposed this internalization process are so extensive that they attain “the kind of universality, invariance, and formal elegance (…) previously accorded only to the laws of physics and mathematics” . Indeed, Roger Shepard’s own work exemplifies how such universality, invariance, and elegance can be achieved in experimental psychology.
Roger N. Shepard is a fellow of the American Association for the Advancement of Science and the American Academy of Arts and Sciences, and is the William James Fellow of the American Psychological Association. In 1977 he was elected to the National Academy of Sciences. In 1995 he received United States’ highest scientific award, the National Medal of Science.
 Castellano MA, Bharucha JJ, Krumhansl CL, Tonal hierarchies in the music of north India. J Exp Psychol Gen 1984;113:394-412.
 Deutsch D, Some new pitch paradoxes and their implications. Philos Trans R Soc Lond B Biol Sci 1992;336:391-7.
 Farrell JE, Shepard RN, Shape, orientation, and apparent rotational motion. J Exp Psychol Hum Percept Perform 1981;7:477-86.
 Georgopoulos AP, Lurito JT, Petrides M, Schwartz AB, Massey JT, Mental rotation of the neuronal population vector. Science 1989;243:234-6.
 Krumhansl CL, Shepard RN, Quantification of the hierarchy of tonal functions within a diatonic context. J Exp Psychol Hum Percept Perform 1979;5:579-94.
 Podgorny P, Shepard RN, Functional representations common to visual perception and imagination. J Exp Psychol Hum Percept Perform 1978;4:21-35.
 Podgorny P, Shepard RN, Distribution of visual attention over space. J Exp Psychol Hum Percept Perform 1983;9:380-93.
 Richter W, Somorjai R, Summers R, Jarmasz M, Menon RS, Gati JS, Georgopoulos AP, Tegeler C, Ugurbil K, Kim SG, Motor area activity during mental rotation studied by time-resolved single-trial fMRI. J Cogn Neurosci 2000;12:310-20.
 Shepard RN, Stimulus and response generalization: A stochastic model relating generalization to distance in psychological space. Psychometrika 1957;22:325-45.
 Shepard RN, Stimulus and response generalization: deduction of the generalization gradient from a trace model. Psychol Rev 1958;65:242-56.
 Shepard RN, Stimulus and response generalization: tests of a model relating generalization to distance in psychological space. J Exp Psychol 1958;55:509-23.
 Shepard RN, The analysis of proximities: Multidimensional scaling with an unknown distance function. I. Psychometrika 1962;27:125-40.
 Shepard RN, The analysis of proximities: Multidimensional scaling with an unknown distance function. II. Psychometrika 1962;27:219-46.
 Shepard RN, Circularity in judgments of relative pitch. Journal of the Acoustical Society of America 1964;36:2346-53.
 Shepard RN, Geometrical approximations to the structure of musical pitch. Psychol Rev 1982;89:305-33.
 Shepard RN, Ecological constraints on internal representation: resonant kinematics of perceiving, imagining, thinking, and dreaming. Psychol Rev 1984;91:417-47.
 Shepard RN, Toward a universal law of generalization for psychological science. Science 1987;237:1317-23.
 Shepard RN, Mind sights. 1990: W.H. Freeman.
 Shepard RN, Perceptual-cognitive universals as reflections of the world. Behav Brain Sci 2001;24:581-601; discussion 52-71.
 Shepard RN, How a cognitive psychologist came to seek universal laws. Psychon Bull Rev 2004;11:1-23.
 Shepard RN, Cooper LA, Mental images and their transformations. 1982, Cambridge: MIT Press.
 Shepard RN, Kilpatrick DW, Cunningham JP, The internal representation of numbers. Cognitive Psychology 1975;7:82-138.
 Shepard RN, Zare SL, Path-guided apparent motion. Science 1983;220:632-4.