61 subjects were given the task of solving a problem having four possible solutions. "Two cords were hung from the ceiling, and were of such length that they reached the floor. One hung near a wall, the other from the center of the room. The subject was told, 'Your problem is to tie the ends of these two strings together.' He soon learned that if he held either cord in his hand he could not reach the other. He was then (...) told that he could use or do anything he wished." A number of objects which might help in the solution of the problem were present in the room. After each solution had been mastered the subject was told to solve the problem in still another way until all of the solutions had been learned. The trials were timed and introspective reports concerning the means of solution were recorded. A number of hints were given from time to time when the subject failed to reach a solution. The author's conclusions are to the effect that "The perception of the solution of a problem is like the perceiving of a hidden figure in a puzzle-picture. In both cases the perception is sudden; there is no conscious intermediate stage; and the relationships of the elements in the final perceptions are different from those which preceded, i.e., changes in meaning are involved." The author feels that trial and error or association by similarity cannot explain his results. (shrink)

Much indirect evidence supports the hypothesis that transformations of mental images are at least in part guided by motor processes, even in the case of images of abstract objects rather than of body parts. For example, rotation may be guided by processes that also prime one to see results of a specific motor action. We directly test the hypothesis by means of a dual-task paradigm in which subjects perform the Cooper-Shepard mental rotation task while executing an unseen motor rotation in (...) a given direction and at a previously learned speed. Four results support the inference that mental rotation relies on motor processes. First, motor rotation that is compatible with mental rotation results in faster times and fewer errors in the imagery task than when the two rotations are incompatible. Second, the angle through which subjects rotate their mental images, and the angle through which they rotate a joystick handle are correlated, but only if the directions of the two rotations are compatible. Third, motor rotation modifies the classical inverted V-shaped mental rotation response time function, favoring the direction of the motor rotation; indeed, in some cases motor rotation even shifts the location of the minimum of this curve in the direction of the motor rotation. Fourth, the preceding effect is sensitive not only to the direction of the motor rotation, but also to the motor speed. A change in the speed of motor rotation can correspondingly slow down or speed up the mental rotation. (shrink)

Ancient Greek mathematics developed the original feature of being deductive mathematics. This article attempts to give a explanation f or this achievement. The focus is on the use of a fixed system of linguistic formulae in Greek mathematical texts. It is shown that the structure of this system was especially adapted for the easy computation of operations of substitution on such formulae, that is, of replacing one element in a fixed formula by another, and it is further argued that such (...) operations of substitution were the main logical tool required by Greek mathematical deduction. The conclusion explains why, assuming the validity of the description above, this historical level is the best explanatory level for the phenomenon of Greek mathematical deduction. (shrink)

This long-awaited work by prominent Harvard psychologist Stephen Kosslyn integrates a twenty-year research program on the nature of high-level vision and mental ...

A productive way to think about imagistic mental models of physical systems is as though they were sources of quasi‐empirical evidence. People depict or imagine events at those points in time when they would experiment with the world if possible. Moreover, just as they would do when observing the world, people induce patterns of behavior from the results depicted in their imaginations. These resulting patterns of behavior can then be cast into symbolic rules to simplify thinking about future problems and (...) to reveal higher order relationships. Using simple gear problems, three experiments explored the occasions of use for, and the inductive transitions between, depictive models and number‐based rules. The first two experiments used the convergent evidence of problem‐solving latencies, hand motions, referential language and error data to document the initial use of a model, the induction of rules from the modeling results, and the fallback to a model when a rule fails. The third experiment explored the intermediate representations that facilitate the induction of rules from depictive models. The strengths and weaknesses of depictive modeling and more analytic systems of reasoning are delineated to motivate the reasons for these transitions. (shrink)

In this article we propose a theoretical framework of distributed representations and a methodology of representational analysis for the study of distributed cognitive tasks—tasks that require the processing of information distributed across the internal mind and the external environment. The basic principle of distributed representations Is that the representational system of a distributed cognitive task is a set of internal and external representations, which together represent the abstract structure of the task. The basic strategy of representational analysis is to decompose (...) the representation of a hierarchical task into its component levels so that the representational properties at each level can be independently examined. The theoretical framework and the methodology are used to analyze the hierarchical structure of the Tower of Hanoi problem. Based on this analysis, four experiments are designed to examine the representational properties of the Tower of Hanoi. Finally, the nature of external representations is discussed. (shrink)