| > Well, anyone who's seen a field of horses knows that "assume any set of n horses has the same color" is not valid. That's not even true! If n is 0 or 1 then the statement is correct, for example. If your response is "It's not true for all n, though." well, the inductive proof doesn't assume it for all n either. It only assumes it for one n at a time. But also, you're getting too hung up on the exact wording of the statement. For example, we could do the same proofs, but put "on farmer Joe's land" onto the statements. If we do that, "anyone who's every seen a field with horses in it" can't tell us if the statement is true or not. Maybe all groups of 2 horses in a field on farmer Joe's land do have the same color, because he sorts his horses. And obviously all groups of 3 horses would have to have the same color, etc. etc. So on farmer Joe's land, A(n=2) is true. And the induction is valid. And using it gives you the correct results! What's the problem with inductive reasoning here? You seem upset that the induction itself might be valid even if n=2 isn't, but I don't know why. The induction is the same no matter whose farm you're on. That's why you need to prove the base case too. > But you're not trying to prove X = 3 and starting out by assuming X = 3. It's really not. There's an entire series of X, and we're saying if we can prove one we can prove the next. You never assume an X when proving that X. And you never assume a higher X when proving a lower X. There are no circles. It's a chain. "Assume X is 3. Then 2*X=6" is just a different way of saying "If X is 3, then 2*X=6" You can rewrite the entire thing without the word assume if you want. You can do the whole thing as "if Y, then Z". "if A(n), then A(n+1)". If pairs of horses are the same color, then triples of horses must be the same color. If triples of horses are the same color, then quads of horses must be the same color. Those sound like reasonable statements to me. Even if we're talking about all horses in the entire world, I could theoretically go kill every non-brown horse, and the original statements would all become true. This is math. You can't object to abstract logic by mentioning real-world facts that could change at any moment. |