Let us rewind the historical tape to 1945, the year in which John von Neumann wrote his celebrated report on the EDVAC (see Chapter 9 ). That same year, George Polya (1887–1985), a professor of mathematics at Stanford University and, like von Neumann, a Hungarian-American, published a slender book bearing the title How to Solve It. Polya’s aim in writing this book was to demonstrate how mathematical problems are really solved. The book focused on the kinds of reasoning that go into making discoveries in mathematics—not just “great” discoveries by “great” mathematicians, but the kind a high school mathematics student might make in solving back-of-the-chapter problems. Polya pointed out that, although a mathematical subject such as Euclidean geometry might seem a rigorous, systematic, deductive science, it is also experimental or inductive. By this he meant that solving mathematical problems involves the same kinds of mental strategies—trial and error, informed guesswork, analogizing, divide and conquer— that attend the empirical or “inductive” sciences. Mathematical problem solving, Polya insisted, involves the use of heuristics—an Anglicization of the Greek heurisko —meaning, to find. Heuristics, as an adjective, means “serving to discover.” We are oft en forced to deploy heuristic reasoning when we have no other options. Heuristic reasoning would not be necessary if we have algorithms to solve our problems; heuristics are summoned in the absence of algorithms. And so we seek analogies between the problem at hand and other, more familiar, situations and use the analogy as a guide to solve our problem, or we split a problem into simpler subproblems in the hope this makes the overall task easier, or we summon experience to bear on the problem and apply actions we had taken before with the reasonable expectation that it may help solve the problem, or we apply rules of thumb that have worked before. The point of heuristics, however, is that they offer promises of solution to certain kinds of problems but there are no guarantees of success. As Polya said, heuristic thinking is never considered as final, but rather is provisional or plausible.
Cognitive Training on the Solving of Mathematical Problems: An EEG Study in Young Men