A true scientist regularly devises experiments to arrive at as accurate an understanding as possible of the question under study. They then draw on these experiments to develop a theory. This thought process is commonly called induction. Like any other scientist, a mathematician also uses experiments—and consequently induction—when making a discovery, whether it is a result, a concept, a method, or something else. You can see this for yourself by performing, on paper or mentally, the Euclidean division of 716,709 by 552. The first digit of the quotient will be 1 ("552 goes into 716 once"), but to find the next digit, you will have to conduct an experiment: "It looks as though it might go three times, but perhaps not; let's try." In fact, 3 is too large, so you will have to try again with 2…
Choosing an experimental protocol ---------------------
But mathematicians also use another type of reasoning—deduction—when proving the statements they have discovered (see an earlier article). As George Polya explains very clearly in his book Mathematics and Plausible Reasoning, mathematicians' experiments are often psychological in nature.
When investigating a problem inductively, a scientist observes the results of experiments before proposing a plausible hypothesis that accords with those observations. First, however, an appropriate experimental design must be chosen. A biologist, for example, must decide which animal species to study, the environment in which the selected animals should be observed, and the conditions under which the observations and experiments will be conducted… Such reflections are essentially psychological in nature. As Polya observes, "In several respects, mathematics is the most appropriate subject for studying inductive reasoning. Such study involves psychological experiments, such as investigating how a hypothesis is strengthened by different kinds of evidence. Because of their inherent simplicity and clarity, mathematical problems lend themselves to this kind of psychological experimentation much better than problems in other fields."
Let us illustrate this idea with a classic, easily understood example. A biologist wants to study reptiles and must therefore investigate their way of life, their habitat, and so on. In short, the biologist must know the animals under study well and, as Polya writes, may even have to "love them."