Is principle of mathematical induction an example of inductive or deductive reasoning?

Question

Is principle 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?

Answer 1

"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.

Answer 2

Proof by induction is indeed inductive. What is the difference between inductive and deductive reasoning? Inductive reasoning starts with a specific case and generalizes it. Deductive reasoning starts with a general concept and makes it more specific.

So what happens in proof by induction? Let's step back before the proof. I'm adding up numbers 1, 2, and 3. I get 6. Okay. 1, 2, 3, 4 and I get 10. Okay. 6 = 3(4)/2. Cool! 10 = 4(5)/2. Oh cool!!

So what if it's just 1? Well, that's just 1(2)/2 = 1. Hey that works. So let's say it works for SOME k. Would that imply that it works for k+1? Yes. Proof omitted for brevity. But what this shows is we can generalize our pattern to all integers greater than or equal to one.

So what did we do? We found a few specific examples. We then showed it worked for a VERY specific example (the base case) and then assumed that it worked for a less specific example, and showed that if it does, then it works for the next example. That then shows us that it works in general. And so we went from specific to general (inductive reasoning).

On the other hand, now that we know that $\sum_{i=1}^{n}{i} = \frac{n(n+1)}{2}$, we can use it to find the specific sum, $\sum_{i=1}^{100}{i}$. Or any other specific case. And that would be deductive reasoning.