THE CONTINUITY-DISCONTINUITY PRINCIPLE AND THE THEORY OF DYNAMIC REPRESENTATIONS
It is time to pool together the key ideas in the last two chapters and combine them into a coherent theory of dynamic representations. I believe that this new theory is not only more consistent to experimental data, but also emblematic of a new avenue of research in the study of human cognition.
5.1 The Representation of Dynamic Motion
By isolating the dynamic information of mass in the stimuli of Experiment 1 and Experiment 2, we have been able to demonstrate that the cognitive system not only represents the mass of an object but also incorporates the representation of mass into the representation of motion and conceives the motion of an object as dynamic.
The subjects in Experiment 1 were found to cognitively distinguish not only a mechanical pattern from a visual pattern but also the rolling patterns of a heavy object from the rolling patterns of a light object. Consequently, the cognitive system was shown to represent dynamic information, different levels of dynamic information, and different mechanical behaviors of objects in correspondence to the different dynamic level of each object.
In Experiment 2, the subjects were further found to represent the mechanical information, i.e., the combination of dynamic information and kinematic or spatiotemporal information, of an object as they cognitively responded to the different levels of mechanical state information of the stimuli. Specifically, the three type of responses, the left-curved, the straight, and the right-curved, were seen to be correlated to the three different levels of mechanical states, heavy-mass/high-speed, light-mass/high-speed, and light-mass/low-speed, respectively. Therefore, the cognitive system conceives a ball of heavy mass at high speed as having a mechanical tendency to continue a curved motion, a ball of light mass at high speed as being free of such a tendency, and a ball of light mass at low speed as having an opposite tendency, i.e., to deviate from its own motion.
These results can be accounted for by the operation of a cognitive continuity constraint on the representation of dynamic motion within the cognitive system, which, upon taking in the information of past motion trajectory of an object, predictably determines its future motion trajectory according to the representational combination between the object's mass and speed, i.e., the representation of mechanical magnitude. In other words, under this constraint, the representation of a stronger mechanical magnitude would be immediately translated in the cognitive system into a representation of a stronger tendency of the ball to continue its own state of motion. Conversely, with the mechanical magnitude represented as extremely weak, no such a tendency would be cognitively attributed to the mechanical state of the ball, i.e., the ball would be represented as rolling more freely, as evidenced by the deviations of the right-curved responses under the low magnitude (i.e., Slow/PP) condition. Since this cognitive constraint essentially determines how much a future trajectory is a continuation of a past trajectory as represented in the cognitive system, it has been termed the Continuity Representation. The Continuity Representation therefore captures more precisely how the cognitive system conceives motion as dynamic by taking consideration of how exactly the conception of mass is incorporated into the conception of motion and explains the conception of the mechanical tendency of an object to continue its mechanical state by the cognitive system as a necessary consequence of the conception of dynamic motion.
Experiment 5 provides additional evidence for the existence of the Continuity Representation in the cognitive system and challenges the traditional explanation for the conception of curved motion. As found in the responses of Experiment 5, an object continues a curved motion only when it has been in a curved motion; in other words, a curved motion can not be imparted from one object to another.
With the Continuity Representation, we therefore find an alternative explanation for the conception of curved motion, which has been so far explained by the Impetus Theory (M. McCloskey, A Caramazza, and B Green, 1980; M. McCloskey, 1983). Following our new findings, the conception of curved motion results from the cognitive system's representation of a dynamic object in an unique motion context and conceiving the motion of the object as being mechanically constrained, according to the mechanical magnitude of the object, to continue or persist in this motion context. Experiment 2 explains exactly how a curved motion is conceived by the cognitive system from its representations of mass and speed and Experiment 5 confirms this new explanation and provides a counterexample to the Impetus Theory. Essentially, we have concluded that the curvedness could not be found in any conception of force or impetus. And, it is only by representing the curved motion context of an object that the cognitive system was found to conceive the object as traveling in a curved path, i.e., conceiving the object as being constrained to continue its own curved motion.
The heart of the Continuity Representation is the cognitive awareness of the continuity or persistence of mechanical motion that a mechanical motion or state of an object can not suddenly disappear or change into a totally different mechanical motion or state. As such, a weaker form of the Continuity Representation, i.e., the Persistence Representation, maintains that, in the conception of a mechanical event, the representation of a motion in the event, like the motion itself, can not suddenly stop existing in the cognitive system. It is with the Persistence Representation that we found the explanation for some of the results in Experiment 3, Experiment 4, and Experiment 5.
In Experiment 3, recall, the disappearance of the horizontal motion of the pendulum bob in some responses produced a straight-down falling path of the bob. However, it has been shown that the representation of the horizontal motion in fact had not disappeared in the cognitive system of the subjects but was only cognitively made latent; as found, it was completely recoverable in these subjects. Similar results have also been found in Experiment 4, where the forward motion had been found recoverable after a backward response or a straight-down response. In both experiments, the recoverability of motion in the responses strongly indicates that the representation of motion in the cognitive system, though it can be made latent or stored away, is necessarily persistent.
Finally, for the representation of dynamic motion, the subjects's sensitivity and responsiveness to individual motions in Experiment 3 and Experiment 4 have led us to the proposal that the different responses in these two experiments are the results of different conceptual combinations in the cognitive system between more basic representations of two or three individual motions in the events. Capturing the essence of the various conceptual combinations is the notion of vector representation, which illustrates how a conceptual combination is cognitively constructed from the individual representations of individual motions; a conceptual combination was shown to have either a sequential structure in time between a salient representation and a latent representation or a simultaneous structure in space between two salient representations. For both structures, however, the basis for a conceptual combination between two or more motion representations was shown to be directly traceable to the cognitive constraint for continuity on dynamic motion, i.e., to the Continuity Representation.
5.2 The Representation of Dynamic Discontinuity
In contrast to the findings about the subjects' representation of dynamic motion, we have also dealt with the cognitive representation of dynamic discontinuities. A dynamic discontinuity is the sudden interruption or the sudden emergence of a dynamic relation between two or more objects. Since, physically, an object generally exists in a certain dynamic context in which it relates to other objects through either dynamic or static forces, a dynamic discontinuity essentially captures the change in the dynamic context of an object.
An acute cognitive sensitivity to a dynamic discontinuity was found through the spatial discontinuities in the responses of the subjects in Experiment 3. These spatial discontinuities in effect illustrate the sudden emergence of a motion in the direction of gravity, reflecting the exaggerated conception in the subjects of the dynamic information in a dynamic discontinuity --- the dynamic action of gravity as the tension of the string suddenly disappeared. The close connection of motion to the dynamic discontinuity as shown in these responses indicates that the cognitive system represents a dynamic discontinuity as mechanically capable of causing a kinetic change in the mechanical state of an object in the direction of the dynamic discontinuity. Particularly, the straight-down responses demonstrate how some subjects had cognitively exaggerated the effect of a dynamic discontinuity to the extreme, even allowing the gravitational motion to completely override the existing horizontal motion.
The significant difference found in Experiment 4 between the representation of the motion of ball A outside the tube and the representation of the motion of ball B outside the tube but traveling in the instantaneous direction of ball A's motion at the impact point also demonstrates the cognitive sensitivity to dynamic content of a dynamic discontinuity and the strong effect thereof. Particularly, ball B's direction of motion indicates that the cognitive system represented the dynamic content of a dynamic discontinuity as a dynamic-push from ball A at precisely the point of impact. The same explanation also applies to Experiment 3a, where the sudden gravitational motion shown in most responses also indicates the representation of a dynamic-push downward as related to gravity. In both experiments, the subjects were therefore found to be very sensitive to such a dynamic-push in a dynamic discontinuity and had over-reacted to it.
The exaggerated effect found for the dynamic content of a dynamic discontinuity forms a sharp contrast to the extremely precise representations by the subjects as to where and when a dynamic discontinuity occurs and the direction of the subsequent motion associated to the dynamic discontinuity. There is no ambiguity in any response that the drawn spatial discontinuity coincides precisely with the dynamic discontinuity in location and the direction of the subsequent motion with the direction of the dynamic discontinuity. In other words, the cognitive system represents with great accuracy the location and direction of a dynamic discontinuity, but it highly exaggerates its dynamic content.
we will use the term Discontinuity Representation to denote the representation of a dynamic discontinuity by the cognitive system.
The cognitive sensitivity to a dynamic discontinuity as captured by a Discontinuity Representation further confirms the general hypothesis that the cognitive system represents dynamic information. With the Discontinuity Representation, we were able to explain the straight-down responses in the pendulum problem, which was in fact first reported by McCloskey (M. McCloskey, A Caramazza, and B Green, 1980; McCloskey,1983), and the qualitative change of motion around the impact in the responses of Experiment 5, which is in fact very consistent to the experimental results of diSessa and White (diSessa and White, 1981). diSSesa and White found that the subjects believed that a new force can wipe out all the previous motion of an object. Similar results can also be found in (Clement, 1982; diSessa, 1982; Viennot, 1979). In addition, what our subjects represented as the dynamic content of a dynamic discontinuity in Experiment 4 in fact closely resembles that of the 'motion-implies-force' conception (Clement, 1982) --- both are the dynamic push-forward of an object in correspondence to the mass and motion of the object. No doubt, it is only by the cognitive representation of this particular form of dynamic information that motion is conceived as 'force' and an impact or dynamic discontinuity is seen as capable of setting another object into motion. It has been found that even six-month old infants represented this particular form of information (A. Leslie and S. Keeble, 1987). It was also shown that, in conceiving such dynamic information in a mechanical impact between two objects, the subjects had conceived the effect of it in relation to the relative mass of each object (J. Todd and W. Warren, 1982). However, it has been shown uniquely that the cognitive system is highly sensitive to this information, identifies it as more salient when it is in a dynamic discontinuity than when it is with a dynamic motion, represents it instantaneously in time (as in Experiment 5), and treats both forms of it (i.e., negative and positive) quite similarly.
5.3 Continuity-Discontinuity Principle
The Discontinuity Representation can be seen in relation to the Continuity Representation, because a Discontinuity Representation always appears in-between two Continuity Representations of two different mechanical states of an object.
Recall, in a Continuity Representation, the emphasis is on how much the motion state of an object is represented as being continuous. In a Discontinuity Representation, the preciseness of the spatiotemporal information and the highly exaggerated dynamic magnitude of a dynamic discontinuity, on the other hand, appear to emphasize exactly the opposite --- the existence of a dynamic discontinuity --- how much it would make a mechanical state discontinuous and change it into a different mechanical state.
Free of dynamic influence from outside, an object persists in and maintains its own unique mechanical state in accordance to its mass and velocity, which is the essence of first law of Newton. The Continuity Representation tunes into the same essential information and interprets the persistence of motion by the continuity constraint on the representation of motion, which therefore can be regarded as the subjective counterpart to the first law of Newton characteristic of human evolution and experience. However, such an experientially and evolutionarily derived representation of motion circumvents the objective and the idealistic concept of free motion characterized by the first law of Newton, incorporates the concept of mass into the concept of motion, and represents a curved motion as likely an special state of motion which is subject to the same constraint for continuity.
On the other hand, under the influence of a newly emerged force from outside, the mechanical state of an object is made to change from its own unique state of motion under the government of the second law of Newton. The Discontinuity Representation emphasizes the presence of an external force by highlighting the space and time of its arrival and by exaggerating its dynamic magnitude and effect on the patient object.
The Continuity Representation and the Discontinuity Representation are, therefore, opposite in the mechanical situation they each separately represents, complementary of each other in information content, and uniquely characterized by two opposite cognitive predilections (i.e., the exaggerated continuity and exaggerated discontinuity of motion). Yet, they fall into a cognitive structure which contains the complete representation of simple mechanical events and help the cognitive system organize the mechanical world into causally meaningful and interconnected segments.
Underlying these two important mechanical representations in the cognitive system are the keen awareness in human cognition about the acting of force and the persistence of motion, though not being represented in the Newtonian terms. Although being qualitatively different from the Newtonian laws in term of quantitative precision, the Continuity Representation and the Discontinuity Representation nevertheless possess some of the key information contained in the first law and the second law of Newton as well as the basic complementary relationship between them. But, while the two Newtonian laws are complementary in term of the Newtonian force, the Continuity and Discontinuity Representations are complementary in term of dynamic discontinuities.
Hence, the Continuity Representation and the Discontinuity Representation are able to help us explain the conceptual connections between different representations in the human cognitive system of different types of mechanical information. As we have shown, they were able to theoretically account for the diverse responses of the subjects under various mechanical conditions. Treated as a whole, they form a general cognitive principle or model for dynamic cognition in a language of Newtonian physics and human information processing. We hereby name it the Continuity-Discontinuity Principle.
5.4 Conceptual Operations under the Continuity-Discontinuity Principle
In the pendulum problem of Experiment 3, a representation of the forward motion can be made latent by the salient representation of the gravitational motion due to the Discontinuity Representation of the dynamic discontinuity; The latent representation of the forward motion can then re-emerge to become a salient representation due to the Persistence Representation of the forward motion.
In the same experiment, two salient representations of the two individual motions were found to be conceptually combined into a salient vector representation, from which various non-straight-down responses were made by the subjects. Representing the two individual motions as equally salient and with reference to a single object, it appears that the cognitive system was forced to combine the two individual representations into one single representation. The spatial discontinuities found in the responses under such a vector representation demonstrate that the conceptual operation of combining the individual representations depends upon the relative strength of each representation, which was shown to be directly related to the Discontinuity Representation of the dynamic discontinuity and the Persistence Representation of dynamic motion.
Experiment 4 brings such a conceptual operation to a wider scope in which the representations of three individual motions were found to be combined. Particularly, we found that a salient vector representation of two individual motions was combined with a latent representation of a single individual motion, which was responsible for a backward response to Task 1 and a forward (i.e., reversed) response to Task 2 of the same subject.
Characteristic of a conceptual operation of combining a salient representation and a latent representation is the sequential ordering or structure of two representations in time; Characteristic of a conceptual operation of combining two salient representations is the simultaneous ordering or structure of two representations in space; a conceptual operation of combining two salient representations and a latent representation then feature both the simultaneous structure and the sequential structure, as in the case of Experiment 4.
Therefore, a conceptual operation of representation combination occurs as the cognitive system organizes the spatiotemporal structure of the Continuity Representations of individual motions and the Discontinuity Representation of a dynamic discontinuity in the event. It is only with the Discontinuity Representation of a dynamic discontinuity that a salient motion representation arises to overcome or make latent an existing Continuity Representation. And, it is only by making a representation of an individual motion latent is the cognitive system able to structure an event in time and combine two representations into a sequential order. Conversely, if both motion representations were salient, they would have to relate to each other in space and be combined into a simultaneous structure.
Therefore, it is by the conceptual operations that the individual representations are organized into specific spatiotemporal structures and across different levels of mental space; We have further found, however, it is the Continuity-Discontinuity Principle which governs these conceptual operations, stipulates the causal bonds between various representations in space and time, and thus describes more definitively the nature of human dynamic representations.
5.5 Information, Force, and Dynamic Representations
The dynamic information in the world can be regarded as the underpinning or raw 'material' from which both the Newtonian dynamic concepts and the human dynamic concepts were developed as the human mind was trying to capture the essence of mechanical events in the world. However, while the Newtonian dynamic concepts are the result of a scientific aspiration and centers therefore on the objective nature of mechanical events, the human dynamic concepts describe and interpret the mechanical events in the world mainly from the perspective of the human perceiver. Specifically, what scientifically capture the dynamic information in the world are the mechanical concepts of force and mass, whose change in space and time exhibits mathematically precise patterns and obeys the mechanical laws of Newton. By contrast, the cognitive system, being exposed to the same dynamic information and having been evolutionarily adapted to it, is endowed with a system of evolutionarily-developed dynamic representations, whose spatiotemporal properties were found to conform to the Continuity-Discontinuity Principle and, as a result, cognitively highlight the causal relationships between objects in a mechanical event. Therefore, while the Newtonian dynamic concepts objectively portray the dynamic information of the world and, as a result, reveal to us the true nature of it, the cognitive dynamic representations are essentially subjective and consequently reveal to us not the nature of the mechanical world but that of the human mind itself.
The Newtonian concept of force essentially details the process of mechanical energy transmission and distribution, objectively delineates the causal connection and progression within a mechanical event, and mathematically simulates mechanical events by embodying dynamic content in its description of the spatiotemporal structures of the events. However, in mathematically characterizing a mechanical event and objectively describing the causal relationships within a mechanical event, the Newtonian concept of force focuses on describing the detailed spatiotemporal change of each mechanical object and leaves out completely the idea of agency so characteristic of human dynamic representations.
Insofar as the Newtonian concepts provide to us an objective and true characterization of the mechanical world, the strong cognitive exaggerations found for the continuity of dynamic motion and the discontinuity of a dynamic discontinuity show that the cognitive system has completely 'reorganized' the mechanical information in the world and conceives a mechanical event from a perspective very different from the Newtonian description of it. Specifically, in sharp contrast to the Newtonian force description, the human dynamic representations appear to de-emphasize the specific spatiotemporal details of motion and, instead, emphasize the causal roles and actions of mechanical objects. For example, to represent a heavy object as more continuous indicates that the cognitive system tends the dynamic identity of the object as it is in motion and consequently submits the representation of motion to the representation of the heavy-object-in-motion; to be cognitively sensitive to a dynamic discontinuity suggests that the cognitive system particularly represents the dynamic 'act' of an object; in addition, to cognitively exaggerate the effect of a dynamic discontinuity shows that the cognitive system is interested more in the action of objects than the spatiotemporal details of the objects as caused by the action; a latent representation of motion particularly demonstrates that the cognitive system treats the spatiotemporal properties as secondary to the dynamic properties in mechanical events.
Therefore, while both are the products of the human mind in its attempt to describe the mechanical information of the world, the Newtonian concepts and the human dynamic concepts describe the world from totally different perspectives, emphasize very different aspects of the mechanical information in the world, and obey different organizational principles in space and time. In contrast to the Newtonian concepts being characteristic of scientific precision and objectivity, the dynamic representations are characteristic of human subjectivity and the processing limitations of the cognitive system. In comparison to the precise mathematical descriptions of the Newtonian mechanics, the cognitive system dismembers the mechanical information in the world and recreates mechanical events in the individual dynamic representations of the components in the events and the combined dynamic representations of the events, not according to complete mechanical veracity but to human predilections and human representation of the mechanical information under the Continuity-Discontinuity Principle. Finally, in correspondence to the Newtonian mechanical laws which govern the spatiotemporal properties of mechanical events, the Continuity-Discontinuity Principle governs the structures of human dynamic representations organized not only in space and time but also between two different representational levels, i.e., the salient and the latent levels, of the human mind.