[Math Lair] Inductive Reasoning

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Inductive reasoning is a type of reasoning that, in contrast with deductive reasoning, which goes from the general to the specific, goes from the specific to the general. Inductive reasoning should not be confused with the principle of mathematical induction; they're two different things with similar names.

Can inductive reasoning lead to new information? This is the problem of induction. What justification do we have to jump from "every raven that has been observed is black" to "all ravens are black?"

In inductive reasoning, all conclusions are, to some degree or another, uncertain. We can only have various degrees of confidence in them. Assuming that inductive reasoning has begun with true premises and followed acceptable principles, it has the property of inductive strength, which means that it is improbable for the premises to be true and the conclusion false.

To illustrate, imagine that there is a certain rule that certain triplets follow, and you have some way of testing a triplet to determine whether it follows the rule or not. Say that the following number triplets follow the rule:

What rule is consistent with this set? Well, if you think about it, you could come up with all kinds of possible rules. Here are a few possibilities:

You can come up with a hypothesis about any of these possibilities; however, no matter how how much you test, is isn't possible for the hypothesis to be "proven" to be true. The relationship between the hypothesis and the results can be expressed logically as if (hypothesis is true) then (certain results are obtained). Concluding that the hypothesis is true because the results obtained were as predicted commits the fallacy of affirming the consequent.

There simply is no pattern of results that can prove a theory true. The best that you can do is to try to disprove as many of the incorrect rules as possible. You would still always have an infinite number of possible rules; in this case it would usually make sense to use the principle of Occam's razor to select the simplest one.

Inductive logic is generally not to be found in mathematical proofs, but that doesn't mean that it is of no use in mathematics. Usually, mathematicians don't sit down, take a bunch of definitions, axioms, and postulates, and start deducing everything possible. Rather, they start by seeing patterns in numbers or geometric figures, which leads them to suspect that some relationship is true, and then they will use deductive reasoning to prove that relationship.