I am in the "I don't understand this but I would like to" camp actually. Your example is exactly why I don't like induction. You have shown that your dog is black by assuming that all dogs are black, which is nonsense.Charlie Reams wrote:That's not really what Nicky said, but anyway.
Consider the following argument (which is not an induction argument): "If all dogs are black then my dog is black." I take it you agree with this statement. One way to prove the statement would be:
1) Assume all dogs are black.
2) Consider my dog. Is he black? By assumption 1, yes.
3) Hence the overall statement "If all dogs are black then my dog is black" is true.
Notice that you agree with the whole statement even though you presumably don't agree with the assumption.
The point of the inductive step is not to establish directly that something holds for all x. It is to establish that, if it holds for any given x then it also holds for x+1. "But what if it doesn't hold for x?", you say. Well, what x would this be? It can't be x=1, because that's the base case which you checked explicitly. It can't be x=2, because by the inductive step, we know that if the statement is true for x=1 (which it is) then it's true for x=2 too. Likewise it can't be x=3, 4, etc... So there can be no case in which the statement doesn't hold; or, to put it another way, the statement holds in all cases. Which is what we were trying to prove.
Incidentally I think you'd have an easier time understanding this if your attitude was "I don't understand this but I would like to" rather than "I don't understand this so it must be wrong."
That's exactly what induction does: I want to show that k(k+1) is positive so they assume that it's always positive from the offset. Similarly you want to show that your dog is black so they assume that all dogs are always black from the offset.