The previous post used the phrase “local data” something like thirty times. The phrase did most of the heavy lifting in the argument: local data does not determine global structure, the more holes the underlying object has the more slack between local and global, and so on. I want to start this post by interrogating the phrase, because once you look at it carefully it turns out to be a beautiful piece of mathematical convention and to be doing far more work than the previous post gave it credit for.
A Word About a Word
“Local” is one of mathematics’ great screwdrivers. It is a tool that works across an enormous variety of contexts, sometimes contexts that look almost nothing alike. We use it for shadow lengths in Egypt, for the curvature of a surface at a point, for the behavior of a function near a value, for individual hiring decisions in a labor market, for pixel values in a CMB sky map, for survey responses in a particular county. The screwdriver does work in all of these contexts. That is the reason we use the same word for all of them. The equivalence the word establishes is part of the conceptual furniture of modern science, and the conceptual furniture is genuinely useful — it lets us say things that would be inconvenient to say without it, and it lets the same arguments travel across fields that otherwise would not be in conversation.
This kind of word is a convention, in the precise mathematical sense. When we say “the usual topology” on the real line, we are not making a metaphysical claim about which topology is correct; we are invoking a shared agreement that lets the conversation proceed efficiently because the ambient context is presumed clear. Conventions are conveniences. They purchase the speed of the conversation by leaving things implicit. They are, in a sense, the Junk Drawer of formal practice: a designated, shared, agreed-upon place where things go that we do not need to specify every time, because we have already specified them well enough that everyone in the room can be expected to remember.
And like any junk drawer, the convention works because we are not constantly reorganizing it. The screwdriver works because we are not asking, every time we use it, whether what we are doing is really screwing. Use the convention, but keep one hand on the receipt would be a fine working slogan for what this post is about. The convention is fine. The convention is necessary. The convention is also, by its nature, partly invisible to the people using it, and the partial invisibility is doing inferential work that no individual practitioner authored or audited.
What “Local” Is Local To
Here is the move that the Bonnet result makes formal. To call data “local” is to say it is taken at a location, and a location is a point in some space, and once you have committed to a space you have committed — at least implicitly — to a global structure that gives the location its meaning. The phrase “local data” pretends to be metaphysically humble (we are only making claims about a small region!) while actually being metaphysically loaded (the small region is a region of something, and the something is doing all the inferential work). The data is not local to a place. The data is local to a manifold, and the manifold is the global object the data is presumed to belong to.
The Bonnet pair is the moment this is exposed. The same point on the donut is “locally” the same as the same point on the Bonnet-paired second donut: same metric, same mean curvature, same everything you can measure at the point. The “locality” of the data — the sense in which the measurement is of this place — is exactly the same in both cases. And yet the data is data about two different surfaces. The data is not local to a place; the data is local to a space, and which space the data belongs to is not something the data tells you.
The Hole, Properly
This puts the hole concept from the previous post on firmer footing. The hole in the donut is defined by the donut as exactly the place where the donut itself is undefined. The donut cannot describe its own complement from within itself, because the donut’s grammar is the grammar of the substance, not the absence. The donut is fully specified at every point of its substance by local data. The hole is what the donut is not. It is the negative feature of the object that constitutes it as the object it is, and it is exactly the place the donut’s local measurements are silent about — because the local measurements are local to the donut, and the hole is the part of space the donut chose not to be.
I want to ask, as the previous post did but with more philosophical traction now, which holes make our world the world it is, and what we can hope to know about them from inside.
The Context Determines the Truth
Before going further I want to say the deepest version of the methodological point this post is going to keep returning to, because all the subsequent moves are versions of it. The natural objection to everything I have just said is something like: “Well, sure, but if the practitioners of a field have done their work — chosen the right context, fixed the convention, audited the framing — then their results within the context are correct, and we should not worry about it.” The response is more disturbing than the objection, and it is the heart of what the post is doing.
The within-context correctness of a convention is not in tension with the of-the-context artifactuality of some of the results the convention generates. They are the same thing seen from inside and outside. The system is doing exactly what a system is supposed to do — generating true statements according to its rules. The fact that some of those statements are partly or wholly artifacts of the rules themselves does not mean the system is making mistakes. It means the system is being itself, and being a system has a constitutive horizon. Within the context, the truths are true. Across contexts, some of those truths are contingent on the context that produced them. And here is the part that disturbs: there is no place inside the system from which one can reliably perceive which is which.
This is Gödel. Or rather, this is the political-economy version of what Gödel showed for formal systems and what Tarski showed for definitions of truth: a sufficiently rich system cannot fully describe its own conditions of correctness from within itself. The Gödel sentence is true, and the system is correct in being unable to prove it, because the system’s correctness is the correctness of its rules, and the rules are what generate the limitation. Gödel is not a story about a system being defective. It is a story about the system being exactly what it is, and that being-what-it-is having a horizon. From outside the system, the horizon is visible. From inside, it is not, and the inside being unable to see it is part of what makes the inside the inside.
Bonnet is the geometric version. The Bonnet pair are two surfaces with identical local data and different global structure, and the local data is correct in both contexts. There is no measurement either donut can make that distinguishes the contexts; the measurements are local to the surface, and the surface is the context. Each donut is correct from inside itself. The fact that there are two of them, and that they are different, is the across-context contingency that neither donut’s measurements can perceive. The hole each donut lives around is exactly the place its measurements are silent about, and the silence is constitutive.
Eratosthenes Was Lucky
It is worth honoring, before going further, that careful inference from data within a well-curated context can sometimes recover global structure. Eratosthenes did it in 240 BCE. Working in Alexandria, he knew that on the summer solstice in Syene (modern Aswan), the sun cast no shadow at noon — light fell straight down a deep well. In Alexandria, on the same day at the same time, a vertical pole cast a shadow at an angle of about seven degrees. From the geometry of the situation, and from the assumption that the sun’s rays arrive parallel, the angular separation between Syene and Alexandria along the surface of the Earth had to be about seven degrees. Multiplying by the known overland distance gave a circumference within a few percent of the modern value.
This is one of the great achievements of human reasoning, and it is worth pausing to honor it. Eratosthenes did not leave Egypt. He did not need a satellite. The Apollo 8 photograph of Earthrise, twenty-two centuries later, did not promote the spherical-Earth hypothesis from “well-supported” to “established.” It removed a particular kind of public-skepticism vulnerability. Eratosthenes already knew.
But Eratosthenes was working in a context whose conventions had been narrowed by generations of pre-Eratosthenian astronomical work. The Greeks had cut the candidate global topologies down to a handful: sphere, infinite flat plane, perhaps a few exotic alternatives nobody took seriously. The shadow at Syene was informative about Earth’s shape only because the hypothesis class had already been curated to one in which the shadow was informative. The screwdriver had been calibrated for his particular paint can. The convention “local data about Earth’s surface” had been settled, by his predecessors, to mean local data about a sphere with not too many other live alternatives, and his measurement was decisive within that settled convention. The within-context truth was close enough to the across-context truth that the gap did not bite.
Eratosthenes was lucky. We are not always going to be Eratosthenes.
Cosmology Is Not So Lucky
Do we know what global topology our universe has? The honest answer is no, and the way the answer is no is a clean illustration of what happens when the convention is operating in a hypothesis class large enough that the within-context vs. across-context gap is wide and known to be wide.
Cosmologists distinguish carefully between the local geometry of the universe — whether space is positively curved, flat, or negatively curved on average — and the global topology, which is the question of how the local pieces fit together at the largest scales. Planck satellite data and the surveys before it indicate that the local geometry is flat to within tight error bars. This does not tell us the global topology. A flat universe could be infinite Euclidean three-space, or a three-torus, or a Bieberbach manifold, or a Picard horn, or a number of other compact alternatives that all satisfy the same local-flatness constraint. People have looked for “matched circles” in the cosmic microwave background — the signature you would see if light from the same point in the early universe reached us along two topologically distinct paths in a non-trivially-connected space — and so far the searches have not found anything definitive.
The most quantitatively privileged field on Earth, working with continent-scale instruments and decades of consensus-driven measurement, has not closed the question of whether we live on a 3-sphere, a 3-torus, or something stranger. The data is doing exactly what it should. The cosmologists know what they are doing. Their convention is fully audited. And the within-context correctness of the cosmological framework is precisely what is unable to settle the across-context question, because the across-context question is exactly the Gödel sentence of the cosmological system: a question whose answer would require occupying a vantage the system cannot occupy from inside. Cosmology has made its peace with this. It works around it. It does not pretend it has gone away.
The Hole Where Time Should Be
So far I have been talking about spatial holes. There is a related kind of hole that I want to flag, because it is one of the genuine reasons political-science questions are so hard. Many of our most important questions — the Lucas critique, Goodhart’s law, every counterfactual policy evaluation, every claim about whether an intervention “worked” — require us to compare the trajectory we got to the trajectory we would have gotten under different conditions. The trajectory we got is a positive feature of our world. The trajectory we would have gotten is exactly a hole — the place our world is not, the part our world chose not to be.
An undergraduate philosophy professor of mine at UNC liked a paper I had written, on something I unhumbly called “a proof of fate,” that made the observation that if you could occupy a higher-dimensional vantage from which a person’s entire life trajectory were laid out as a single object — the way a Family Circus cartoon shows Billy’s loopy dotted-line path through a day as a single static figure — then the trajectory exists as an object whether or not Billy experiences himself as choosing at each moment. The cartoonist’s view is the four-dimensional view. Billy does not have it. We do not have it for ourselves. From inside Billy’s day, the trajectory we are not on is a hole. The cartoonist sees the trajectory; Billy sees only the location he is at.
Some of the most powerful methodological developments of the last several decades — Pearl’s causal inference, the do-calculus, the entire industry of DAG-based identification — are partial recoveries of the cartoonist’s view from inside Billy’s. The trick, in a phrase, is to commit to a higher-dimensional structure (the causal graph) that the data does not by itself give you, and to use that structure to read the lower-dimensional data correctly. Simpson’s paradox dissolves once you commit to a DAG. The marginal and conditional distributions stop fighting once you have the cartoon of the trajectory rather than just Billy’s sequence of locations. I plan to spend more time on this in coming posts, because the trick is genuine — but it is not free, and the price is the price the rest of this post has been telling you about. The DAG is a context. Within it, the inferences are correct. Across DAGs, the inferences are contingent. The convention, the receipt, the screwdriver, all over again.
The Active Version
The political-science version of all of this is sharper than the cosmological one in a way worth being honest about. Arrow and Phelps showed in the early 1970s that an employer with a prior belief that women are on average less productive will, in equilibrium, see exactly that — because the prior lowers the returns to costly skill acquisition for women, fewer women invest, and the resulting hire pool validates the prior. The same local datum (“this candidate did not invest in costly skill acquisition”) is fully consistent with two genuinely different global hypotheses: a labor market in which women are less productive, and a labor market in which women rationally decline to invest because the market discriminates. No amount of additional measurement at the level of individual candidates resolves the question, because the equilibrium population is itself a function of the prior.
Within the context of the equilibrium that has formed, the employer’s belief is correct. The hires are, on average, less qualified than they would have been otherwise. The data confirms the prior. The prior is doing exactly what a prior should do. And the equilibrium that has formed is one of multiple equilibria the underlying productivity distribution would have been consistent with. Across contexts, the labor market could have been somewhere else, with different practices, different priors, and different — better — outcomes. The within-context correctness of the employer’s belief is not in tension with the of-the-context contingency of the equilibrium; they are the same thing viewed from inside and outside. The employer is not making a mistake. The system is being itself. And the system being itself is partially constituting the very thing it is correctly describing. Maggie Penn and I have been working on the formal generalization of this for some years now, in the language of classifier-feedback dynamics; the recent paper is the rigorous version of an old idea, and the old idea is exactly the active form of the topological underdetermination problem the previous post identified.
The cosmological case is informationally hard but causally inert: the universe does not adjust its topology in response to our beliefs about it. The political case is informationally hard and causally entangled: the prior over the global structure is partially making the global structure that determines what the local data means. The hole is not just there. The hole is being maintained, in part, by the way we are looking at it. The convention by which we look has become part of the structure we are looking at, and the within-context correctness of the looking is part of what is keeping the structure where it is.
The Convention and the Junk Drawer
The Junk Drawer arc on this blog has been about taxonomies that work locally and fail globally — categories that are useful, shared, and silently constitutive of the practices that use them. The “local data” convention is a taxonomy at one level of abstraction up. It is a category that lets us treat shadow lengths and pixel values and individual hiring decisions as the same kind of thing, because the word “local” travels across them. The travel is doing inferential work. The inferential work is invisible from inside any of the practices the convention serves, because each practice is using the convention correctly within its own context. None of them is wrong. And the convention as a whole is doing something none of them authored.
This is the meta-theme I think this blog has been circling around for a long time without having quite said directly. Most of the formal apparatus we use — vote-counting, classification, measurement, regression, identification, even the language of “local” itself — comes with conventions that the practitioners rarely articulate, and the conventions are doing far more work than the formalism gives them credit for. The conventions are not wrong. They are conventions. They are correct within the contexts they were designed to serve. They are also partially constitutive of what counts as a result inside those contexts, and they are silent about the holes — the places the contexts are not, the equilibria that did not happen, the trajectories that were not taken, the topologies the data cannot distinguish.
The Shape of the World From Where We Stand
Copernicus did not visit the sun. Eratosthenes did not leave Egypt. Pearl did not stand outside time. The victories these figures won were inferential victories, paid for by hypothesis classes carefully chosen and conventions carefully defended. They are real victories, and the project of inferring global structure from data taken inside the structure is not hopeless. It is also not innocent. The convention is doing as much work as the data, and pretending otherwise is the methodological version of the mistake Bonnet’s intuition made.
The world we are doing political science about has more holes than the world Eratosthenes was doing geography about. Some of those holes are spatial — institutions that fail to fit together at scale, federal-state-local seams that local data cannot resolve. Some are temporal — counterfactuals we cannot run, regimes whose data was generated under conditions we cannot recover. Some are epistemic — priors whose validation is being maintained by the very practices the priors authorize. None of these holes is going to show up in our measurements. By the formulation we started with, they cannot. They are exactly what our measurements are not about. And the conventions we use to take the measurements are partly what is keeping the holes where they are.
The honest version of the project, the one I think MOP keeps coming back to, is something like this: use the convention, but keep one hand on the receipt. The convention is what makes the conversation possible. The convention is also what makes the silences invisible. Within the convention, our results are true. Some of them are also partly artifacts of the convention itself, and from inside the convention we cannot reliably tell which is which. The polity is its silences as much as it is its substance. We are not Eratosthenes. The system we live inside has Gödel sentences the system itself cannot perceive, and some of them are being held in place by the system continuing to be the system.
With that, I leave you with this.