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In mathematics, you can define things in different ways to get different answers. Ways of defining things tend to be highlighted as true (in at least some conte
by dkarl 3y ago
In mathematics, you can define things in different ways to get different answers. Ways of defining things tend to be highlighted as true (in at least some context) if they are interesting and useful, and ignored if not. I don't think the definition based on "dividing into pairs" is particularly interesting or useful in the context of the child's understanding of numbers, because it's too vague to be useful, and it doesn't lead to any insights.
The definition based on transfinite ordinals explained in the same answer does seem interesting, and I wouldn't be surprised if it were useful. I think this is a case of simplification gone wrong, where everything interesting was lost in the translation to more accessible terminology.
A more honest thing to say to a child would be that the way even and odd are defined only make sense for finite numbers. It's true for the definition they know, and it introduces them to the important insight that logical rules that are created for one kind of thing might not work when applied to something else. I think this would be more accessible and stimulating for a six-year-old than giving them a half-baked verbal imitation of a result from transfinite mathematics.
They'll be thrilled later if they study math and discover that there are definitions of "infinity" and "even" that yield an answer to their childhood question.
- amelius 3y agoAn even more honest thing to say is that infinity when used as a number is a hack introduced by mathematicians to make notation and reasoning simple in some cases, but that it can be dangerous in other cases, like any other hack. If you want to use infinity in a safe way, then use limits around your expressions. (And this quickly resolves the case of this article, since lim x->inf x-2*floor(x/2) does not exist).
- dkarl 3y agoThat's true in calculus, and probably a lot of other applied mathematics contexts where rigor tends to get swept under the rug, but it's not completely fair since there are versions of "infinity" that are defined and used rigorously. (The transfinite ordinals and cardinals mentioned in the Stack Overflow article are the example I'm familiar with.)
- chowells 3y agoIt's not a hack to create a new set and work out rules for how to use it which are both internally consistent and support easy morphisms with more familiar sets. It may not be easy, but it's hardly a hack. It's one of the big ways math works, really. Are negative numbers a hack? Rational numbers? Algebraic numbers? Well then neither is the two-point compactification of the reals or extending the natural numbers into the ordinal numbers. These are things with very precise models and interpretations. No hacks at all.
- amelius 3y agoBut nobody said that hacks cannot have precise interpretations. It's the unreasonable cognitive load that is the problem.