Geometry: Inductive and Deductive Reasoning
Inductive reasoning is the process of arriving at a conclusion based on a set of observations. In itself, it is not a valid method of proof. Just because a person observes a number of situations in which a pattern exists doesn't mean that that pattern is true for all situations. For example, after seeing many people outside walking their dogs, one may observe that every dog that is a poodle is being walked by an elderly person. The person observing this pattern could inductively reason that poodles are owned exclusively by elderly people. This is by no means a method of proof for such a suspicion; in fact, in the real world it is a means by which people and things are stereotyped. A hypothesis based on inductive reasoning, can, however, lead to a more careful study of a situation. By inductive reasoning, in the example above, a viewer has formed a hypothesis that poodles are owned exclusively by elderly people. The observer could then conduct a more formal study based on this hypothesis and conclude that his hypothesis was either right, wrong, or only partially wrong.
Inductive reasoning is used in geometry in a similar way. One might observe that in a few given rectangles, the diagonals are congruent. The observer could inductively reason that in all rectangles, the diagonals are congruent. Although we know this fact to be generally true, the observer hasn't proved it through his limited observations. However, he could prove his hypothesis using other means (which we'll learn later) and come out with a theorem (a proven statement). In this case, as in many others, inductive reasoning led to a suspicion, or more specifically, a hypothesis, that ended up being true.
The power of inductive reasoning, then, doesn't lie in its ability to prove mathematical statements. In fact, inductive reasoning can never be used to provide proofs. Instead, inductive reasoning is valuable because it allows us to form ideas about groups of things in real life. In geometry, inductive reasoning helps us organize what we observe into succinct geometric hypotheses that we can prove using other, more reliable methods. Whether we know it or not, the process of inductive reasoning almost always is the way we form ideas about things. Once those ideas form, we can systematically determine (using formal proofs) whether our initial ideas were right, wrong, or somewhere in between.