Conceptual Modeling in physics, mathematics and cognitive science

Scientific thinking is grounded in the evolved human ability to freely create and manipulate mental models in the imagination. This modeling ability enabled early humans to navigate the natural world and cope with challenges to survival. Then it drove the design and use of tools to shape and control the environment. Spoken language facilitated the sharing of mental models in cooperative activities like hunting and in maintaining tribal memory through storytelling. The evolution of culture accelerated with the invention of written language, which enabled creation of powerful symbolic systems and tools to think with. That includes deliberate design of mathematical tools that are essential for physics and engineering . A mental model coordinated with a symbolic representation is called a conceptual model. Conceptual models provide symbolic expressions with meaning. This essay proposes a Modeling Theory of cognitive structure and process. Basic definitions, principles and conclusions are offered. Supporting evidence from the various cognitive sciences is sampled. The theory provides the foundation for a science pedagogy called Modeling Instruction, which has been widely applied with documented success and recognized most recently with an Excellence in Physics Education award from the American Physical Society. The Copernican Revolution in science culminated in Newton‘s Principia (1687), which integrated astronomy and terrestrial physics into a single science of motion. Immanuel Kant (1787) saw this as a striking union of mathematical theory with empirical fact that bridged the traditional divide between rationalism and empiricism. So he proposed a comparable “Copernican Revolution” in philosophy to account for it [1]. Just as Copernicus shifted the center of the universe from earth to sun, Kant shifted the focus of epistemology from structure of the external world to structure of mind. His revolutionary insight was that our perceptions and thoughts are shaped by inherent structure of our minds. He argued that the fundamental laws of nature, like the truths of mathematics, are knowable precisely because they do not describe the world as it really is but rather prescribe the structure of the world as we experience it. Though the scientific revolution has expanded in spectacular fashion to integrate physics and astronomy with chemistry and biology, Kant’s revolution in philosophy has hardly progressed. His profound influence on the epistemology of physics is evident in the writings of Einstein and Bohr as well as many other scientists and philosophers. However, continued debates on such topics as the interpretation of quantum mechanics show no signs of consensus, and they have overlooked recent advances in cognitive science with high relevance to epistemology. My purpose here is to open a new stage in Kant’s revolution by explaining how findings of cognitive science can be marshaled to create a new “science of mind” with testable predictions and explanations as required of any “true” science. I begin with a restatement of Kant’s primary question: What does the structure of science and mathematics tell us about how the human mind works? In searching for answers my working hypothesis will be: The primary cognitive activities in science and mathematics involve making, validating and applying conceptual models! In a word, science and mathematics are about MODELING –– making and using models! This essay argues for a “MODELING THEORY of MIND” to guide the multifarious branches of cognitive science in research on the nature of mind and brain, and the design of conceptual tools for science and mathematics. Core principles are explained and supporting evidence is sketched, but the brush is necessarily broad. More details are given in [2,3,4], especially for application to physics teaching and learning. I. NEWTON’S MODELING GAME Newton did much more than provide the first mathematical formulation of a scientific theory in his Principia; he also demonstrated how to relate it to empirical fact. Though Kant recognized revolutionary implications for epistemology in this impressive feat, physicists have overlooked it. The issue has been thoroughly explicated in [5] by framing Newtonian theory in terms of models and modeling, so brief mention of key points is sufficient here. Newton could not make the crucial distinction between model and theory explicit in his original formulation, because the concept of model did not emerge in scientific discourse until the nineteenth century. But [5] shows that he made it implicitly. The point is that theoretical principles like Newton’s Laws cannot be tested or applied except by incorporating them in models. Thus, models mediate between theory and experiment. And Newton’s Laws can be regarded as a system of design principles for making models to describe, to predict, to explain and to control motions of material bodies. Kant’s insight can be explicated by noting that Newton linked up two distinct kinds of models: theoretical and empirical. A theoretical model derived from Newton’s Laws predicts motions, while an empirical model derived from data describes a motion. A match between them explains a motion. In this way Newton explained Galileo’s law of falling bodies and Kepler’s three laws of planetary motion. Note the distinction between a theoretical Law (with a capital L) and an empirical law (with a lowercase l), also called an empirical model. Comparison between theoretical and empirical models is such a standard practice of physicists since Newton that they seldom consider its profound epistemological implications. At its simplest, it involves creating an empirical model from data with a procedure often called “curve fitting.” That’s how Kepler’s laws were derived. It is an important technique in the search for empirical regularities that are both quantifiable and reproducible. In high energy physics data analysis has become so complex that a new research specialty has emerged to handle it. That research, often called “phenomenology,” is thus intermediate between theory and experiment. For future analysis, it is worth noting that scientific work in all three domains is governed by definite but different rules; from mathematical rules for theorists, to measurement standards for experimentalists, to probability theory for phenomenologists. As Kant recognized, scientific objectivity requires strict adherence to rules. The question is: Where do the rules come from? II. FROM COMMON SENSE TO SCIENTIFIC THINKING As we grow and learn through everyday experience, each of us develops a system of common sense (CS) concepts about how the world works. To evaluate introductory physics instruction, the Force Concept Inventory (FCI) was developed to detect differences (in student thinking) between CS concepts and Newtonian concepts about motion and its causes [6]. Results from applying the FCI were stunning from the get-go! First, the differences were huge before instruction. Second, the change was small after instruction. Third, results were independent of the instructor’s experience, teaching method and peer evaluation. These results have been replicated thousands of times from high school to Harvard and in 25 different languages. The FCI is by far the most cited reference in the physics education literature, and it is widely used today to evaluate the effectiveness of teaching reforms. Here we are interested in what the FCI tells us about human cognition. The FCI is based on a taxonomy of 35 CS concepts in 5 major categories [6]. These concepts are overlooked or summarily dismissed as misconceptions by most physicists. However, they are common outcomes from everyday experience, and they are quite serviceable for dealing with physical objects. Moreover, central CS concepts in the 5 categories have been clearly articulated and discussed by major intellects of the preNewtonian age, including Newton himself before the Principia [7]. So CS concepts should be regarded as alternative hypotheses about the physical world that, when clearly formulated, can be tested empirically. For example, the CS concept: “a moving object implies existence of a force (a mover)” contravenes Newton’s First Law. The Second Law is contravened by the concept that forces are due to “active agents” (usually living things), so there are no passive forces, although motion is deflected by passive objects called “barriers.” The Third Law is contravened by the common metaphorical notion that