Vanishing sets and ideals · 11:52pm Jul 5th, 2022
This is my third time trying to write this post. The previous two times, I failed to find a way to write about this well, so I'll instead write about it badly.
I started trying to understand algebraic geometry (very) recently, and I bumped into what's called the Nullstellensatz. I haven't understood it yet, but there's a slice of the intuition that I found fascinating.
The Nullstellensatz is about vanishing sets and ideals. In math, if you want to prove something, you need a way to represent (1) the thing you want to talk about, and (2) the thing you want to prove, both with mathematical objects. Algebraic geometry represents things as the zeros of polynomials. Intuitively, if you want to talk about Ponies, you find a way to describe all the ways an object can fail to be a Pony, and you know you have a Pony when you're unable to fail the object. The set of all objects that meet all Pony Criteria is referred to as a vanishing set. If you set up your Pony Criteria correctly, then the corresponding vanishing set will be the same as the set of all ponies.
You can also imagine some other Criteria to keep track of the loose ends in a story. The vanishing set, in that case, might be the story's resolution after the climax, after which there are no more loose ends.
Some kinds of criteria are very special in math. They're the ones that deal with structure. They describe objects that only look at how things get pieced together, and they don't care which things exactly are getting pieced together. For example, a pair describes two of anything, and it doesn't matter which things. If you have two of them, then you have a pair of them. Similarly, a sequence describes an ordered collection of things, and it doesn't matter which things. You can have more specific structures, like a “frankenstein” (I made that up). Let’s say a frankenstein pieces together body parts into some monster, and it doesn’t matter which body parts as long as you have the right number of the right types (head, arms, legs, and so on).
In a sense, structure describes when our weird propensity to see patterns in things is actually a logical thing to do. If something looks like it has a certain structure, then it does have that structure. (Caveat: You can’t just eyeball it in math. You need to be precise about the requirements.)
The criteria for structures, at least for the kinds of objects the Nullstellensatz deals with, is called an ideal. An ideal is a collection of criteria that:
- Behaves as you would expect criteria to behave. Meaning, if X and Y are both part of the criteria, then so is the combination XY of both. (For polynomials, it’s “closed under addition and subtraction.”)
- And still applies even if you transform the criteria to make it show up in other scenarios, so long as you don’t mangle the relationships between its different parts. (For polynomials, it “absorbs multiplication.”)
If you have any structure, you can imagine that it has a corresponding ideal, which gives you a checklist of all the ways some object can fail to have that structure. Then the vanishing set of that ideal gives you all possible objects that have that structure.
Intuitively, you probably have some abstract conception of particular themes that stories can have. Those themes don’t care too much about which stories are involved, as long as it has the right sorts of story elements coming together in the right sorts of ways. In a sense, each theme describe some structure that a story can have. An ideal for a theme would be way of representing the theme that makes it easy to find when a story fails to uphold that theme. The vanishing set of that ideal would be the set of all stories that uphold the theme.
In these terms, the, uh… let’s call it the Literary Nullstellensatz… tells us something about the way we see themes in stories. The idea is that we don’t just look at stories in isolation, we look at them as if they were made up of many smaller, sometimes overlapping, stories, and each of those mini stories can have their own themes. By looking at stories that satisfy a given theme and then the themes that those stories all satisfy, we can start to recognize all of these mini stories and the roles they play in piecing together the bigger theme.
Anyway. This is the third time I’ve failed to write this blog post. I intended to have one more paragraph about the role played by the structure of seeing one thing in another, which was my original motivation for writing this post. Unfortunately, I don’t even know how to write that one badly. Maybe I’ll try again once I actually understand the proof of the Nullstellensatz.
Thanks for reading.
Vanishing set is like pathetic fallacy in that the adjective just baffles me. How does the set vanish? What's pathetic about the fallacy?
Is it simply a mistranslation, the way begging the question is a mistranslation of petitio principo which itself is a mistranslation of the Greek phrase (honestly, no wonder everybody misuses it)?
5670399
Agreed. It has other names, including variety, which I think is much more appropriate. I ended up going with vanishing set because I thought it would be easier to understand when I introduced the concept and because it makes for a good reminder of how ideals are represented. I might have misjudged.
I'm actually very fond of plays on words that sound weird and border on absurd, but make perfect sense when treating words algebraically. It looks like pathetic fallacy is one of those. I think it's supposed to be a-apathetic, where the double-a cancels out. I use the words "logic" and "analogic" in a similar way: analogical reasoning is like logical reason but where the usual relationships for deduction go backwards. It happens to be the case that pointing the arrows backwards often aligns well with reasoning-by-analogy, so usually I don't need to expound when talking through my thoughts with my own weird definition.
5670399
The expression "vanishing set" is similar to "boiling point". What they mean is not a set that vanishes or a point that boils; it's the set where something vanishes and the point where something boils.
5670456
You're alive! Welcome back.
I still think of you as the person that basically taught me math, by the way. I hope you're not too embarrassed by this blog post.
This discussion on structure reminds me of one of the more frustrating elements of my day job: it's far harder to convince someone that two things are similar enough to fit into the same category than to show the same person that two similar things are, in fact, different enough to be placed in different boxes.
5670592
That's a cool observation. It seems related to the problem of abstraction, where you want to reference multiple things interchangeably, so you represent those things with a single name/variable and don't further specify which one you're talking about. I think usually abstraction is the goal when using structure, rather than the underlying objects, to reason.
Maybe people are reluctant to put two things into the same category when they don't understand why it's important or useful to abstract those two things.
Y'all are too smart for me.
5670624
I think it's also an issue of people who primarily think in terms of deductive vs. inductive reasoning. The P/J distinction of MBTI may be similar. Any sufficiently-trained monkey can correctly perform deductive logic. However, that sufficient training often teaches students to ignore (quite possibly extraneous) patterns in favor of starting from first principles.
Deductive logic gone wrong is best summed up by "a madman is someone who has lost everything but his reason"; inductive logic gone wrong is schizophrenia (or, in a less severe case, children picking up on indicental commonalities in the training data and then drawing seemingly bizarre conclusions when tested with novel examples and vanilla-grade superstitions in adults).
5670533
Thank you :)