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Mathematical induction

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Despite its name, mathematical induction is not truly inductive. It is deductive. We are said to use inductive reasoning when we say that, because we have seen a billion black ravens, it must be the case that all ravens are black. Inductive reasoning is probabalistic, but does not provide formal certainty or necessity. This is opposed to deductive reasoning, in which it is impossible for the premises to be true and the conclusion false.

There is a certain sense in which mathematical induction is similar to the probabalistic induction previously described. As Suber notes,

"it generalizes to a whole class from a similar sample...the sample is usually a sample of one, and the class is usually infinite. Mathematical induction is deductive, however, because the sample plus a rule about the unexamined cases actually gives us information about every member of the class. Hence the conclusion of a mathematical induction does not contain more information than was latent in the premises. Mathematical inductions therefore conclude with deductive certainty"(Suber)

Suppose we want to determine whether or not every even number is divisble by 2. But how can we definitively determine this? It's not as though anyone has ever examined every single even number in existence. Indeed, this is impossible, because there are an infinite number of even numbers. It is this inference concerning unexamined cases from examined cases that makes mathematical induction (superficially) similar to the more popularly known probabalistic induction.

The reason why the even numbers are decisively different from coffee shop burgers lies in the logic of mathematical induction. We can prove that the smallest even number (namely, 2) is divisible by 2. This is our very small sample. And we can prove that the next even number after every number divisible by 2 will also be divisible by 2. This is our rule about the unexamined cases. That is enough to imply that the successor of 2, namely 4, will be divisible by 2, and its successor, 6, and its successor 8... and so on ad infinitum. This is how a small sample and a rule about unexamined cases can give us information about every case. This is how our knowledge of an infinite set of unexamined cases can be as certain as the conclusion of a valid deduction, quite unlike the conclusion of an ordinary induction(Suber).

First, we demonstrate that a theorem holds for a "minimal case"(Suber). This is our smallest sample size; namely, the number 2. We can provide a proof that "if [the rule] is true of the sample, then it is true of the unexamined members of a class"(Suber). It therefore follows that the theorem holds for every member of a specific class (Suber).

Suber, Peter. "Mathematical Induction." Retrieved from: http://legacy.earlham.edu/~peters/courses/logsys/math-ind.htm

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