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Is mathematical induction a deductive or inductive argument?

  1. Nov 26, 2009 #1
    1. The problem statement, all variables and given/known data

    Is mathematical induction a deductive or inductive argument?
    Would appreciate the help. Thanks.

    Jeremy

    2. Relevant equations



    3. The attempt at a solution
    Its name suggests that the process is inductive, yet I know all of mathematics depends on deductive reasoning...
     
  2. jcsd
  3. Nov 26, 2009 #2
    Short answer: Your hunch was dead on. It's deductive reasoning (inductive is not accepted as a valid type of reasoning in most disciplines).

    Long answer: The first step in mathematical induction is inductive. I'll use as an example the formula for summing all consecutive integers from 1 to n:

    [tex] 1 + 2 + ... + n = \frac{n(n+1)}{2}[/tex]

    The first step in proving this is to prove that it's true for n = 1. That is:

    [tex] 1 = \frac{2}{2} = 1[/tex]

    To stop there would be to use inductive reasoning - i.e., since it's true for n = 1, it must be true for all n. This is obviously not necessarily correct, and that's where the deductive part comes in. The purpose of a deductive argument is to prove that, given a hypothesis, its conclusion must be valid and follow directly from the hypothesis. That is, now that we know that the above is true for n = 1, we assume that it's true for some n (that's the hypothesis), and show that it must then be true for n + 1. Now that it's in general form like that, you've completed the deduction, and shown that it's true for all n in the domain of the problem (in this case, natural numbers).
     
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