spotlighted concepts

by paying the blog tax, i unlock the ability to not have to avoid this platonic concept due to worries that no one will have any idea what i'm on about

There’s this term, a verb “to spotlight” or a participle adjective “spotlighted”, which is used a few different times in Planecrash:

If you’ve actually got to negotiate with very humanlike aliens, you need Financial Literacy Layer 5; or at least, Keltham hopes that’s what he needs, because that’s what he has. This gives him access to a spotlighted permutation-based method for determining the fair contribution of one actor to a multiagent process. It’s not spotlighted nearly as hard as, say, the Probability axioms, or Validity; but it’s pretty much the only spotlighted method for that kind of fairness, and Civilization is somewhat hopeful that aliens will use it too.

Combine this with simpler ideas like “If you prefer 100% of one thing to 100% of another, you should prefer higher probabilities of getting that thing rather than the other, in gambles between them” and you can pretty thoroughly spotlight the Law of Probable Utility showing that Preference() must compound probabilities with utilities the same way that probabilities compound with each other.

It’s basically being used in the figurative “to draw attention to” sense, which isn’t novel. But, in my own head if nowhere else, the connotations of the word as used in these examples are a little more specific than that. And even though it’s only used in this specific way in like, four tags, this concept has become a small part of my internal conceptual vocabulary, when thinking about things like the process of mathematical abstraction.

I am not quite sure if it’s a concept that clicked quite so hard to other people who read Planecrash, and “offhand mentions in a small handful of scattered paragraphs throughout a million word glowfiction” is a terrible way for a concept to be documented anyways, so I figured it was worth trying to articulate the concept that ended up in my head.

There will be math, because I am too busy trying to write thirty articles in thirty days while still making progress on other projects to make a point of not adding math. Sorry!

A mathematical object, structure, definition, or so on is spotlighted if it lies at the bottom of some sort of attractor basin within the space of mathematical concepts. Spotlighted concepts are the ones that we might expect aliens to find, especially when the reasons we might expect aliens to find them have more to do with potentially-surprising deep structure of mathematics. Showing that many apparently unrelated avenues identify the same object spotlights that object. Theorems that identify an object as the unique object satisfying some natural property or list of properties spotlight that object, especially when there are several distinct such theorems.

Let’s consider an example. Imagine little April, in sixth grade or whatever, learning about compound interest the first time. Maybe this weird number called e comes up.

Suppose we ask sixth grader April how much she thinks aliens care about the number e. What does she think? Well, if she’s just hearing of it now for the first time, looking at that limit, maybe she isn’t convinced there’s much that’s special about this number. In actuality, probably she’s gotten the impression through osmosis of mathematical discussion that e is somehow important—isn’t there something called a natural logarithm? But the whole limit thingy isn’t really helping. Why should aliens care about that expression?

Now, suppose you’ve gotten ahold of high school freshman April, who (nearing the end of the school year) has a reasonable if basic idea of what a derivative is. And you go in and tell her

Theorem 1.0: Let f: R → R be a function satisfying f’(x) = f(x). Then f is of the form

where A is any real number. Since r⁰ = 1, we necessarily have A = f(0). As a corollary, the derivative of a function given by b^•(x) = b^x is

Now, maybe you relate to a statement like this a little differently, if you haven’t been to college and haven’t really gotten the formal concept of a function¹¹ A function is, of course, a cobijective binary relation. down solidly, if you’re only just now being introduced to the terms “domain” and “range” and you haven’t yet really had to make a habit of very clearly differentiating the “image” from the “codomain”.

But I think it’s more impressive than the limit thing. I can readily imagine that the aliens might care about functions which are their own derivative, and I can also imagine that after figuring out the power rule for differentiating functions like x², they might next ask about functions like 2^x.

Similarly, the common statement

does not seem to me to spotlight e all that hard, on its own? Little April does not understand why the value of e^iπ + 1 is supposed to be of note, it seems like a sort of arbitrary expression.

Now, start expanding it out into

and explaining what’s going on with it, and then we’re getting somewhere. An incredibly convenient algebraic expression for converting between polar and rectangular coordinates?

But maybe don’t be too hasty, there—how much does this actually spotlight e in particular? Consider the value

This value has the remarkable property that

where cos_r and sin_r give the coordinates of a point which has gone around a unit circle θ times, rather than θ /τ times.

So, how convincingly this spotlights the real number e is going to depend on how much you care about our standard cos and sin functions in particular, compared to other sinusoids that represent the same things, but which don’t express angles in radians²² My favorite way to spotlight radians is the observation that sin and cos are their own fourth derivatives, which only happens when you take the input in radians, because you want (cos(θ), sin(θ)) to walk around one unit around the unit circle for every unit you change θ by.

It’s a pretty convincing spotlighting, even though it frustrates the part of me that thinks “revolutions” is obviously the most natural unit to measure angles in..

Ever wonder why anyone cares about complex conjugation? Here’s a spotlight pointing at it:

Theorem 1.1: Other than the identity, the topological field C has precisely one continuous automorphism (structure-preserving bijection), given by

This automorphism is also the only automorphism φ such that φ(R) = R.

If you’re literally just considering the abstract field (C, +, *), then there actually isn’t any way to determine which elements are reals. But this is silly—we generally³³ Normally, you put the order topology (ordered fields can only be ordered in one way, because “which elements have square roots” is enough to sort between positive and negative) on Q to get R, and then take the algebraic closure. But if you take the algebraic closure of Q and then give it a (nondiscrete) topology and complete it, you get a set isomorphic to C, but we haven’t nailed down enough details for R to be recoverable. Something basically like this can happen with the p-adics.

(And Q is just the integers with division—but if you want to be fancy, it’s an initial object in the fields of characteristic zero.) make C by taking R (the unique ordered field with supremums⁴⁴ I think something close to this is the only good characterization of R you're going to find. I guess you could say like, it’s the only complete topological field that can embed into any complete field where the subspace topology of Q is equivalent to Q’s order topology, or something. But that condition is probably just going to end up being equivalent to the supremum thing.) and adding a square root of a negative number, or taking an algebraic closure, or something like that. And when we do this, we don’t usually want to forget which elements are the old ones and which are the new ones!

The reals have no automorphisms other than the identity, so that’s fixed in place. And then you add, say, a square root of minus one. But if you just define i = √(-1)… -i also satisfies that definition, it’s also a square root of minus one. You can swap the sign of all your i's and -i’s, and, as long as you make sure you hit all of them, nothing at all changes⁵⁵ See also this cursed chicanery.. And because i and -i are interchangeable in this sense, it’s not too surprising that a+bi and a-bi play nice together in a lot of ways too.

This spotlighting thing is why it makes sense to care about “coherence theorems”, like Cox’s theorem or Dutch book theorems or whatever.

Why do probability theory in the particular way we do? Well, Cox’s theorem says that any way to quantify uncertainty that satisfies a certain set of coherence conditions which all seem pretty reasonable⁶⁶ Quantum mechanics does lead me to wonder if asking uncertainties to be real numbers is a mistake. See also this lecture, where Scott Aaronson tries to spotlight some aspects of quantum mechanics. will end up being isomorphic to standard probability theory⁷⁷ Though note that this isn’t an argument against using log odds..

Why model agents as maximizing the expectation of some utility function? Well, Dutch book theorems argue that you’ll lose out on money if you don’t, and the seemingly-reasonable VNM axioms necessitate it. Now, bounded rationality gets complicated⁸⁸ Citation needed, of course, and I would recommend taking care to avoid using mathematical arguments like this to justify or rationalize foolish behavior, but the clear and simple spotlighted model will still, to an appreciable extent, reflect part of the structure of the more messy Reality.