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Is principal of mathematical induction an example of inductive or deductive reasoning according to wikipedia it says it is deductive as it is just a mathematical proof but according to the definition of inductive reasoning it should be inductive instead of deductive. Can someone help?

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    $\begingroup$ What is the difference between induction and deduction? $\endgroup$ – parsiad Oct 9 '16 at 15:45
  • $\begingroup$ Induction means going particular to general and deduction means general to particular. $\endgroup$ – Matt Oct 9 '16 at 15:48
  • $\begingroup$ I'm not sure what that means, but I would recommend, since I assume this is for the purpose of building intuition, to pick whichever speaks best to you. $\endgroup$ – parsiad Oct 9 '16 at 15:48
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"Proof by induction," despite the name, is deductive. The reason is that proof by induction does not simply involve "going from many specific cases to the general case." Instead, in order for proof by induction to work, we need a deductive proof that each specific case implies the next specific case. Mathematical induction is not philosophical induction.

It might be helpful here to recall in detail what proof by induction means (there are variants, such as transfinite induction, but let's ignore them for now). Induction states that:

If $P$ is some property of natural numbers, and $(i)$ $P$ holds of $1$, and $(ii)$ for every $n$, if $P$ holds of $n$ then $P$ holds of $n+1$; then $P$ holds for all natural numbers.

Note that in order to use this, we need to prove both parts. It's not enough to observe "$P$ holds of $1$, $P$ holds of $2$, $P$ holds of $3$, therefore $P$ always holds;" the whole "meat" of induction is the induction step, which is the part where you prove that $P(n)$ implies $P(n+1)$ for every $n$.

If you look at an example of proof by induction done in detail, you'll see what I mean.

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  • $\begingroup$ Thanks so much for such a simple answer it helped. $\endgroup$ – Matt Oct 9 '16 at 16:07
  • $\begingroup$ @RaghavSingal Glad to help! $\endgroup$ – Noah Schweber Oct 9 '16 at 16:09

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