This year I'm attempting to teach myself and relearn a ton of math, so resurrecting the blog (again) to think things through seems natural. Future posts will come as need and opportunity arise.
The blog has neither comments nor RSS, (both broke years ago) so I'm probably talking to myself. Also, math is currently in HTML, which is inadequate, but I don't want to find a better solution now.
I'm working on teaching myself Real Analysis from a set of course notes and the book Principles of Mathematical Analysis by Rudin. The course notes start by defining the natural numbers and addition, taking the Well-Ordering Principle as an axiom, and deriving the Principle of Math Induction. It goes on to recursively define multiplication and exponentiation on the natural numbers, and to prove the properties of all three operations using induction. OK, so far so good.
The notes then go on to prove the formulas for ∑i=1→n ip for various values of p. The first step is to derive the formula. Start with np+1 = ∑i=1→n ip+1 − ∑i=1→n−1 ip+1. Do a bunch of algebra and the right side turns into a linear combination of sums from 1 to n of powers ≤p. You can then recursively determine a closed form solution for the sum of ip.
The notes assert that this is not sufficient to prove that the formula is valid for all n. In order to prove it, you must use mathematical induction. However, the notes don't say anything about why the derivation is not a valid proof (or why it could not be rearranged to become a valid proof). It feels to me like all the math in the derivation is correct, so it should be sufficient as a proof. I can't tell if the derivation is sufficient, but the proof by induction is there as practice in proof by induction. Another possibility is that there is some mathematical weakness in the derivation, so the proof by induction is necessary. If there is a weakness, it is not obvious to me, and the notes do not explain.
The book does not help at all. It starts with the rational numbers, and I don't think it has anything to say on the subject of induction.
This bumps into my general nervousness around math proofs. If asked to prove something, I'm always comfortable jumping in and playing with the math and working something out, but I'm often unsure of what starting assumptions I can use. Over and over I write something and ask, "is this actually valid? I mean, I know it is, but do I have to prove it before I can use it?" In this case, I don't know why the derivation is not a valid proof, but I can accept that. But for future proofs, I don't know if I will know what the standards are.