This paper explores the classical idea of complementarity in mathematics concerning the relationship of intuition and axiomatic proof. Section I illustrates the basic concepts of the paper, while Section II presents opposing accounts of intuitionist and axiomatic approaches to mathematics. Section III analyzes one of Einstein's lecture on the topic and Section IV examines an application of the issues in mathematics and science education. Section V discusses the idea of complementarity by examining one of Zeno's paradoxes. This is followed by (...) presenting a few more programmatic suggestions and a brief summary. (shrink)

Einstein learned from the magnet and conductor thought experiments how to use field transformation laws to extend the covariance to Maxwell’s electrodynamics. If he persisted in his use of this device, he would have found that the theory cleaves into two Galilean covariant parts, each with different field transformation laws. The tension between the two parts reflects a failure not mentioned by Einstein: that the relativity of motion manifested by observables in the magnet and conductor thought experiment does not extend (...) to all observables in electrodynamics. An examination of Ritz’s work shows that Einstein’s early view could not have coincided with Ritz’s on an emission theory of light, but only with that of a conveniently reconstructed Ritz. One Ritz-like emission theory, attributed by Pauli to Ritz, proves to be a natural extension of the Galilean covariant part of Maxwell’s theory that happens also to accommodate the magnet and conductor thought experiment. Einstein's famous chasing a light beam thought experiment fails as an objection to an ether-based, electrodynamical theory of light. However it would allow Einstein to formulate his general objections to all emission theories of light in a very sharp form. Einstein found two well known experimental results of 18th and19th century optics compelling (Fizeau’s experiment, stellar aberration), while the accomplished Michelson-Morley experiment played no memorable role. I suggest they owe their importance to their providing a direct experimental grounding for Lorentz’ local time, the precursor of Einstein’s relativity of simultaneity, and do it essentially independently of electrodynamical theory. I attribute Einstein’s success to his determination to implement a principle of relativity in electrodynamics, but I urge that we not invest this stubbornness with any mystical prescience. (shrink)

The Computational Metaphor is an extremely influential notion, and more than any other trend has given rise to the field of Cognitive Science. Environmentalism is at present better formalised as a political movement than as a scientific paradigm, despite significant research by Gibson and his followers. This article attempts to address the difficult problem of synthesising these two apparently antagonistic research paradigms.

The book presents the case that cognitive science should turn its attention to developing theories of human cognition that cover the full range of human perceptual, cognitive, and action phenomena. Cognitive science has now produced a massive number of high-quality regularities with many microtheories that reveal important mechanisms. The need for integration is pressing and will continue to increase. Equally important, cognitive science now has the theoretical concepts and tools to support serious attempts at unified theories. The argument is made (...) entirely by presenting an exemplar unified theory of cognition both to show what a real unified theory would be like and to provide convincing evidence that such theories are feasible. The exemplar is SOAR, a cognitive architecture, which is realized as a software system. After a detailed discussion of the architecture and its properties, with its relation to the constraints on cognition in the real world and to existing ideas in cognitive science, SOAR is used as theory for a wide range of cognitive phenomena: immediate responses ; discrete motor skills ; memory and learning ; problem solving ; language ; and development. The treatments vary in depth and adequacy, but they clearly reveal a single, highly specific, operational theory that works over the entire range of human cognition, SOAR is presented as an exemplar unified theory, not as the sole candidate. Cognitive science is not ready yet for a single theory – there must be multiple attempts. But cognitive science must begin to work toward such unified theories. (shrink)

On the basis of a comparative analysis of the biosemiotic work of Jakob von Uexküll and of various theories on biological holism, this article takes a look at the question: what is the status of a semiotic approach in respect to a holistic one? The period from 1920 to 1940 was the peak-time of holistic theories, despite the fact that agreement on a unified and accepted set of holistic ideas was never reached. A variety of holisms, dependent on the cultural (...) and disciplinary contexts, is sketched here from the works of Jan Smuts, Adolf Meyer-Abich, John Scott Haldane, Kurt Goldstein, Alfred North Whitehead and Wolfgang Köhler. In contrast with his contemporary holists, who used the model of an organism as a unifying explanatory tool for all levels of reality, Jakob von Uexküll confined himself to disciplinary organicism by extending the borders of the definition of “organism” without any intention to surpass the borders of biology itself. The comparison reveals also a significant difference in the perspectives of Uexküll and his contemporary holists, a difference between a view from a subjective centre in contrast with an all-encompassing structural view. Uexküll’s theories are fairly near to J. S. Haldane’s interpretation of an organism as a coordinative centre, but even here their models do not coincide. Although biosemiotics and holistic biology have different theoretical starting points and research-goals, it is possible nonetheless to place them under one and the same doctrinal roof. (shrink)

In our view, the way of thinking involved in insight problem solving is very close to the process involved in the understanding of an utterance, when a misunderstanding occurs. In this case, a more appropriate meaning has to be selected to resolve the misunderstanding , the default interpretation has to be dropped in order to “restructure”, to grasp another meaning which appears more relevant to the context and the speaker's intention. A new conception of unconscious, implicit thought emerges, informed by (...) relevance. In this article we support our view with experimental evidence, focusing on how a misunderstanding is formed. We have explored two problems, in which a trivial arithmetical task is represented as an insight problem and vice versa. Studying how an insight problem is formed, and not just how it is solved, may well become an important topic in the contemporary debate on thought. (shrink)

Summary Why and how is the Gestalt theorists’ concept of productive thinking particularly suitable for being applied to the educational question of how student motivation can be encouraged, thus providing an important condition for self-regulated, intrinsically motivated learning? An answer to this question has been sought using an approach to the fostering of text comprehension ability, based upon the features specific to productive thinking, originally identified by Wertheimer and Duncker. Firstly, these specific features are dealt with and their educational implications (...) compared with those deriving from the definitions of problem-solving used most frequently in educational research. Secondly, an analysis is made of the process by which the features specific to productive thinking are turned into the conditions for a kind of text analysis suitable for designing an instructional project aimed at enhancing text comprehension ability and, at the same time, encouraging intrinsic motivation and self-regulation on the part of the learner. Thirdly, an educational project centred on the thinking-aloud poor reader is described, where thinking aloud and reflection–response are combined in order to guarantee the maximum level of intrinsic motivation. In the concluding section, the most important features of the project are discussed in relation to reciprocal teaching and scaffolding. (shrink)

Max Wertheimer, in his classic Productive Thinking, linked understanding to transfer: Understanding is important because it provides the ability to generalize the solution of one problem to apply to another. Recent work in human and machine learning has led to the development of a new class of generalization mechanism, called here analysis‐based generalization, which can be used to provide a concrete account of the linkage Wertheimer suggested: these mechanisms all, in different ways, use understanding of examples in the generalization process. (...) In this paper I review this class of mechanism, and describe a method for causal attribution that can produce the analyses of examples that the generalization methods require, in the domain of simple procedures in human‐computer interaction. This causal analysis method is linked with analysis‐based generalization to form EXPL, an implemented model which is a concrete, though limited, instontiation of Wertheimer s scheme. EXPL constructs an understanding of an example procedure and generalizes it on the basis of that understanding. Results of an empirical study suggest that some of EXPL's attribution heuristics are used by people, and that while a subclass of analysis‐based methods, called superstitious methods, seem to provide a more plausible account of people's generalization under the conditions of the study than a contrasting class of rationalistic methods, at least some participants appear to use methods from both classes. The results also show that explanation‐based methods, which rely on comprehensive domain theories, must be used in conjunction with a means for extending the domain theory. If thus enhanced, explanation‐based methods are able to mimic the effects of other analysis‐based methods, and can provide a good account of the data, though combinations of other methods must also be considered. Finally, I return to Wertheimer s ideas to argue that none of the current analysis‐based generalization methods fully captures Wertheimer s notion of understanding. Proper choice among different possible analyses of an example is crucial for Wertheimer, but I argue that this problem may be beyond the reach of learning systems. (shrink)

The richness of humanity is the diversity of its cultures, but now as never before the destructive power of modern technology and threatening ecological disasters make it necessary that we all recognize we are many peoples of one world. Complementing the diversity of our different cultures, the growth of a common, scientific knowledge inspires the hope that we may achieve and share a secondary culture of ideas. Computers, which can help represent explicitly the best ideas of modern science, can aid (...) in the diffusion of such powerful ideas to create a popular, secondary, scientific culture.We propose that a primary objective of learning environment design should be the development of thinkable models in the minds of students; further, since thinkable models represent a level of knowledge deeper than superficial terminology and specific graphical forms, such may be an objective suitable for various people in different cultures. To pursue these notions in some detail, a taxonomy of models is developed and the issue of how representations relate to human modes of perception and action is raised. The notions are explored first through the contrasting of a half-dozen approaches to the Pythagorean Theorem; then through describing polylingual word worlds and the Rosetta disk project.Common experiences produce common models —not only public models but the cognitive structures built from those shared experiences; and shared models will permit enhanced communication and understanding. If sharing experiences through play with computer based learning environments in different languages permits children to develop common models of the world, this will ultimately enhance mutual understanding between people in different places. This objective is now within reach. (shrink)

The purpose of the two studies reported here was to develop an integrated model of the scientific reasoning process. Subjects were placed in a simulated scientific discovery context by first teaching them how to use an electronic device and then asking them to discover how a hitherto unencountered function worked. To do this task, subjects had to formulate hypotheses based on their prior knowledge, conduct experiments, and evaluate the results of their experiments. In the first study, using 20 adult subjects, (...) we identified two main strategies that subjects used to generate new hypotheses. One strategy was to search memory and the other was to generalize from the results of previous experiments. We described the former group as searching an hypothesis space, and the latter as searching an experiment space. In a second study, with 10 adults, we investigated how subjects search the hypothesis space by instructing them to state all the hypotheses that they could think of prior to conducting any experiments. Following this phase, subjects were then allowed to conduct experiments. Subjects who could not think of the correct rule in the hypothesis generation phase discovered the correct rule only by generalizing from the results of experiments in the experimental phase.Both studies provide support for the view that scientific reasoning can be characterized as search in two problem spaces. By extending Simon and Lea's (1974) Generalized Rule Inducer, we present a general model of Scientific Discovery as Dual Search (SDDS) that shows how search in two problem spaces (an hypothesis space and an experiment space) shapes hypothesis generation, experimental design, and the evaluation of hypotheses. The model also shows how these processes interact with each other. Finally, we interpret earlier findings about the psychology of scientific reasoning in terms of the SDDS model. (shrink)