Classification - gradients and undecidability
A random thought I had recently. My algebra professor was telling us about how classification is often a key question in mathematics. I.e. classification of finite simple groups. Or whether it’s possible to tell whether two groups are isomorphic.
And I thought about how it’s a very human thing to try to do (Ok, no, he mentioned it too, but I generalized). We try to classify species in biology, we try to classify people in all sorts of ways, e.g. MBTI types. Or gender/race/nationality/etc. (I’ve always thought that aliens would have a lot of the same math as we have. And I still do think so... but I wonder if there would be a focus on classification like that)
And how we’re really not good at it. We always seem to find that no, stuff doesn’t fit into neat buckets, it’s a sliding scale. We find problems with all the various buckets, they’re imperfect. Gender being an obvious one that’s a talking point now.
Or rather, it’s not that we’re not good at it, it’s just not the way the world works, trying to classify things is just fundamentally a sort of error.
And this extends backwards into math too. Our classification problems are often literally unanswerable, in general you can’t decide whether two groups are isomorphic! You look at two things, and you can’t even figure out if they’re the same, perhaps can’t even figure out which of your buckets they fit into. (I’m not sure, as an example, whether determining properties like nilpotency is decidable.) It’s in a way a wrong question.
Are there classifications more like a gradient in mathematics? I can think of some that are in some abstract way like that, maybe. The kernel of a morphism is a measure of how close it is to being a monomorphism. And yet usually the question is “is it a monomorphism”, from what I understand, black and white. Are ‘fuzzy classifications’ a common thing, when complete classification is undecidable?
