Comments by "EebstertheGreat" (@EebstertheGreat) on "How to Add" video.

  1. I have a bunch of miscellaneous notes about this video. First, at least once, Roof (sorry if I forgot your name) makes the mistake of using 1 as the base case instead of 0 (he points this out in a text overlay). This is an easy mistake to make, because in fact, the natural numbers are sometimes defined to exclude zero (and were originally defined this way). Whether to call the positive integers or the nonnegative integers the "natural numbers" has long been a matter of disagreement, and even today many textbooks do not include 0 in the natural numbers. In these books, base cases are written out with 1, but otherwise there is no meaningful difference. It does eventually lead to slightly different wording for things like the Fundamental Theorem of Arithmetic, but basically there is no difference. The motivation for doing it that way is probably continuity with Peano's axioms in their original form, though I also saw a Real Analysis book that seemed to take that approach just because of the elegant way it allowed it to write the definition of natural numbers and succession: Let 0 := Ø, N be an infinite set (called the set of natural numbers ) such that 0 ∉ N and there exists a bijection S: {0} ∪ N → N . Remarkably, that is all you need. Or maybe it's not that remarkable, because it does make sense that the only structure one needs for counting is that each number be followed by a new number. And the only thing that makes 0 special is that it has no predecessor. (0's other properties show up in the definitions of addition and multiplication.) A serious downside of this approach is that to be totally rigorous, one needs to prove the existence of a Dedekind-infinite set (i.e. you have to prove that there exists such a set N and map S with the stated properties, which is not the case in some theories), but the book ignored that, because it was above the target 300-level undergrad course. Second, the principle of induction cannot be avoided in any of these proofs, because it is fundamental in every respect. Addition and multiplication are defined inductively to start with. The natural numbers themselves are defined inductively. (Roof did not actually give a definition of the natural numbers in any video so far, but roughly speaking, they are all and only the numbers that can be reached by using the successor on 0 over and over again. A more rigorous definition is kind of subtle, which is why I like the simple one I gave above.) At the highest level, the process of induction is usually justified these days by well-ordering, though it can equivalently be justified by well-foundedness. The definition I gave above is well-founded with respect to succession, because every natural number is a successor of something, but 0 is not the successor of anything, so every natural number reduces to S(S(...S(0)...)), with some finite number of applications of S(). Proofs that use the Well-Ordering Principle are arguably circular, since the proof of the Well-Ordering Principle itself uses a form of mathematical induction (that relies on well-foundedness). At any rate, suffice it to say that there are many ways to prove this principle applies to the natural numbers, and to the ordinal numbers, and even to sets ordered by inclusion. Third, I think the waffling over the predecessor is sort of unnecessary. All you have to say is that "every number is either 0 or the successor or some number, so I'll define this for 0 and then I'll define it for successors." That's the reason there are two parts to each definition anyway. Fourth, if we really wanted to be pedantic, there are an almost endless number of other details we could dig into. Arithmetic is complicated. For instance, how do we know that if 3 = S(2), that means 7×3 = 7×S(2)? It's clearly correct, but how do we prove it? We need some general principle of substitution into equality, which is not exactly hard to prove, but it is another case where intuition can certainly get in the way. And what about numbers 10 and larger? If we really wanted to be strict about it, we would have to go through all the work of defining positional notation and proving various properties about it just to even introduce these numbers. Interestingly, one can use this to rigorously prove the convergence and correctness all of the algorithms used in grade school for decimals (or for numbers in arbitrary bases), like long multiplication and division. I don't think I have ever seen a book go through the painstaking detail of all of these steps (except the dreaded Principia Mathematica ), but in principle, they are all there, in math proof heaven, making up the underlying structure of arithmetic.
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