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Sniffnoy comments on How to Not Lose an Argument - Less Wrong
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Sorry, I can't figure out what you mean here. Of course you can't rigorously prove something that's not true.
I have a feeling that our conversation boils down to the following:
Me: There exists a case where induction fails at n=2.
You: For all cases, if induction doesn’t fail at n=2, doesn’t mean induction doesn’t fail. Conversely, if induction fails, it doesn’t mean it fails at n=2. You have to carefully look at why and where it fails instead of defaulting to “it works at n=2, therefore it works.”
Is that correct, or am I misinterpreting?
Anyways, let's suppose you're making a valid point. Do you think that my interlocutors were arguing this very point? Or do you think they were arguing to put me back in my place, like TheOtherDave suggests, or that there was a similar human issue that had nothing to do with the actual argument?
To butt in, I doubt your interlocutors were attempting to argue this point; they seem like they were having more fundamental issues. But your original argument does seem to be a bit confused.
Induction fails here because the inductive step fails at n=2. The inductive step happens to be true for n>2, but it is not true in general, hence the induction is invalid. The point is, rather than "you have to check n=2" or something similar, all that's going on here is that you have to check that your inductive step is actually valid. Which here means checking that you didn't sneak in any assumptions about n being sufficiently large. What's missing is not additional parts to the induction beyond base case and inductive step, what's missing is part of the proof of the inductive step.