Modeling the Physical World with Common Sense and Mathematics

Physics and mathematics are grounded in man’s evolved ability to freely create mental models and use them to manage interactions with the natural world. 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. 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. A central point is that theoretical principles like Newton’s Laws cannot be tested or applied except by incorporating them in models. 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).