We are now halfway through our book. It may appear to be a coincidence that insight and creative thinking appear at the “center” of a book on problem solving, a coincidence that the gestalt psychologists would surely have liked. Contemplating how the young Gauss solved an arithmetic problem, and considering how a mutilated checkerboard problem is solved, while realizing that the solution of these problems are analogous to how snowflakes look, may come as a surprise. It should not because all three are based on symmetry and invariance. Before discussing how “ordinary” insight problems are solved, this chapter describes the insights that led Galileo, Archimedes, and Einstein to their scientific discoveries. Physicists have known for over a century that there would be no science based on the natural laws, and no natural laws in the first place, if there were no symmetry in nature. So what encouraged this to happen just over a 100 years ago? In 1918, Emmy Noether formulated and proved her mathematical theorems that revolutionized physics. Her theorems showed how the conservation laws can be derived from the symmetry of these laws by applying a least-action principle. The review of symmetry in scientific discovery presented in this chapter provides the stage for a new formalism of problem solving that may apply not only to the sophisticated areas of science, but also to “ordinary” brain teasers, as well as to the TSP and the 15-puzzle.

Popper and the logical empiricists focused on the logical status of the products of research and made scientific discovery and invention, the processes of achieving creative breakthroughs, exogenous to the logic of science. Creative insights were the result of happy but accidental psychological experiences, of minimal cognitive interest. Kuhn, in Structure, attempted to endogenize discovery, to provide an account of the practice of scientific problem-solving but without employing traditional logic of discovery or justification. Key to his account of normal science was the role of exemplars (standardized problem solutions) and acquired resemblance relations (analogies, metaphors, similes). Kuhn’s account was insightful in suggesting that puzzle solving amounts to rhetoric-based problem reduction to existing exemplars, largely independent of theory reduction. While Kuhn helped to resurrect philosophical interest in the process of research, he was only partially successful. By taming discovery in normal science, he exacerbated his problem of understanding how revolutionary breaks are conceived. And he left us with several questions about exemplars. What, exactly, are exemplars and where do they come from? Can exemplars carry across revolutionary breaks? I employ examples from early quantum theory.