In 1867, the French mathematician Pierre Ossian Bonnet asked a question that sounds like it should have an obvious answer. If you know the metric of a compact surface at every point — the intrinsic distances and angles, the things you could measure if you were a tiny ant walking on the surface — and you also know the mean curvature at every point — the extent to which the surface is bending in the three-dimensional space it sits in — does that determine the surface?
The intuition is that the answer is yes. You have continuous, perfect, complete local information. The global shape ought to be pinned down. Bonnet proved this for spheres in his original paper. The case of compact surfaces with at least one hole — tori, in the simplest version — has been open since. Most people who thought about it expected the same answer.
Last week, the news cycle picked up a paper by Alexander Bobenko (TU Berlin), Tim Hoffmann (TU Munich), and Andrew Sageman-Furnas (NC State), published last fall in Publications Mathématiques de l’IHÉS, that resolves the compact case. The answer is no.
Their construction is explicit. They build a pair of analytic tori — closed, doughnut-shaped surfaces — sitting inside ordinary three-dimensional Euclidean space, with the property that the metric is identical at every point of both surfaces and the mean curvature is identical at every point of both surfaces. The two surfaces are not congruent. Their global shapes are different. You can know everything Bonnet asked about and still not know the surface.
Spheres are uniquely determined by their local data. Tori are not. The difference is the hole. Once a closed surface has a hole, there is slack between what local data can tell you and what the surface actually is, and the slack is large enough to fit two genuinely different shapes inside. The machinery in Bobenko-Hoffmann-Sageman-Furnas suggests that these “Bonnet pairs” exist in considerable variety once the genus is at least one. The slack is not a fluke. It is structural.
I want to spend the rest of this post on what this means for the rest of us.
Mathematics has just stated, in the cleanest possible setting — continuous, deterministic, fully measurable, with no noise and no agency anywhere in the system — a principle that political science keeps rediscovering the hard way. Local information does not determine global structure. The conditional distribution does not determine the marginal. The pairwise comparison does not determine the ranking. The arrest rate per stop does not determine who gets stopped. The classifier rule applied at every point does not determine who ends up in the population, because the population is shaping itself around the rule.
Simpson’s paradox is the version of the principle most political scientists meet first. You can know the relationship between two variables inside every subgroup of the population without knowing the relationship across the whole population. The conditional distributions can be fully specified and the marginal can still flip on you. We have an entire series on this — incarceration, the CPI, tariffs — and the unifying claim of all three posts is essentially “Bonnet’s intuition was wrong.”
Arrow’s theorem is the deeper version. Every pairwise majority preference among voters can be known, and known to be locally well-defined, and yet there can fail to exist any global ordering of alternatives that respects all of them. The pairwise comparisons are the local data; the social ordering is the global object. Local agreement at every pair of points does not produce coherent global structure. The slack is there, and the topology of the space of permitted preference profiles determines how much slack you get. Three implies chaos for the same reason that genus one implies indeterminacy: the underlying object has enough structure to permit multiple consistent global continuations of the same local data.
Maggie Penn and I have been working for several years on classifier feedback dynamics, and the formal version of that work — “Classification Algorithms and Social Outcomes” — has just appeared in print at the American Journal of Political Science after a long pipeline. (Maggie wrote a short non-technical summary for the AJPS blog when the paper went online; the formal version is now between covers.) The piece is structurally the same problem as Bonnet’s, at one further remove. A classifier is a local rule. Its application at every point in the population is fully specified. What the population looks like, however, is not determined by the rule alone, because the population is responding strategically to the rule. There can be multiple equilibrium populations consistent with the same classifier behaving the same way at every point. The local information is the rule. The global object is the equilibrium population. The slack between them is the entire substantive content of the problem.
What unites these examples is a single mathematical observation, now sharpened by the resolution of the Bonnet problem: the more holes the underlying structure has, the more slack there is between local and global. A sphere has no holes; its local data determine its global shape. A torus has one hole; its local data do not. Higher-genus surfaces presumably have more slack still, although the differential-geometric question for them is technically still open.
There is a deeper principle here that I want to flag rather than fully develop, because it deserves its own arc. The question Bonnet asked — can a being who lives on a surface determine the surface from measurements made on the surface? — is the question Edwin Abbott asked in Flatland: can a two-dimensional being know that it inhabits a two-dimensional world? It is the question that runs through the physics-of-political-networks series I started earlier this year: do our representational choices about how to draw a network commit us to global structure that is not visible in the local data? It is, eventually, the question Gödel asked about formal systems and the question Tarski asked about truth: can a system describe itself completely from within itself? The shape of the limitation differs case by case. The fact of the limitation does not. We choose how to represent the world, but we can only express the world within the world itself, and there are things about the world that cannot be measured, formulated, or proved from inside it.
In political and social systems, the “holes” are not topological in the literal sense. They are the structural complications that make a system more than the sum of its local pieces — strategic agency, aggregation across heterogeneous groups, endogenous response to measurement, cyclic preferences, feedback loops between observation and observed. Each of these introduces a hole, in the relevant analogical sense. Each of these introduces slack between what we can observe locally and what is true globally. Each of these makes the system harder to pin down by piling on more local data.
This is, I think, the cleanest way to state the conservation-of-impossibility theme that runs through most of MOP. Impossibility results are not solved by collecting more local information. They are relocated. If your local measurements at every point are consistent with multiple global structures — and in any system with enough structure to be interesting, they generally are — then no amount of additional local measurement will resolve the indeterminacy. The slack lives somewhere, and you have to either accept it, change the question, or pay attention to the global object directly.
Bobenko, Hoffmann, and Sageman-Furnas have given the rest of us a clean, non-controversial example to point at when this comes up. Two doughnuts, identical at every point, genuinely different overall. No noise. No error. No agency. No aggregation. Just topology and geometry, behaving the way they always behave when there is a hole in the structure.
The political science version is messier. The doughnuts do not push back when you measure them. Populations do. Voters do. Markets do. Legal classifiers do. The same structural slack is at the bottom of all of it, and the same lesson follows. Local data is not the same as global truth, and treating them as the same is a category error that 150 years of differential geometry could not quite get away with.
With that, I leave you with this.