Democracy, we like to think, runs on ideals like freedom, equality, the consent of the governed. But look under the hood and you'll find something less lofty – mathematics.
The idea that democracy depends on math is not new. From ancient Greece onward, votes had to be counted and seats had to be divided. Mathematics even shaped America’s founding. The Declaration of Independence is, at its core, a mathematical document. It opens with axioms—that all people possess certain unalienable rights—and then asserts a theorem: the British rule must end. And because no theorem is complete without a proof, the Declaration supplies one: "To prove this, let Facts be submitted to a candid world."
This was no accident. Jefferson and Adams studied Euclid and admired the formal architecture of the Elements. Eleven years later, the dominant influence at the Constitutional Convention was the Enlightenment, with mathematical and scientific reasoning at its heart.
Mathematics still underlies nearly every interaction we have with democracy. Tallying votes, allocating legislative seats, sizing legislatures, and drawing district maps all rely on complex mathematical ideas. More importantly, mathematics can tell us whether these mechanisms are fair, whether they produce meaningful representation and incentivize cooperation over polarization and gridlock.
But if the marriage of mathematics and democracy is old, the democracy it now serves is new. We live amid intense polarization, collapsing trust in institutions, litigation as political warfare, and eroding norms. Recent setbacks for voting rights and fair districting underscore democracy’s fragility.
These developments are changing not only the practice of democracy but the mathematics it demands. And mathematics is responding, aided by data science, computer science, political science, economics, and law, and by computational power that makes large-scale modeling possible.
New research on voting, for example, analyzes real-world data to reveal how voting methods behave in practice. Ranked choice voting (RCV), in which voters rank candidates in order of preference, emerges as superior to the plurality or “pick one” method used in most elections. Drawing on thousands of ranked choice elections from the United States, Scotland, and Australia, researchers have shown that RCV elects broadly supported candidates while reducing vote-splitting, spoiler, and strategic voting. Simulations calibrated to real survey data confirm RCV’s advantage even when voters abstain, cast partial ballots, or misjudge the candidates’ ideological position.
Recent research also scrutinizes single-winner districts, whose problems run deeper than the current gerrymandering wars. Massachusetts, for example, elects an all-Democrat Congressional delegation, even though some 30% of its voters are Republican but are spread so thinly that no district can elect one of their own no matter how the lines are drawn. Mathematical modeling shows multi-member districts, which are larger and elect several representatives at once, would fix this. Three Massachusetts districts electing three representatives each would let Republicans win about three seats, matching proportionality that most of the world’s democracies treat as a pillar of their electoral systems.
Remarkably, multi-member districts with a proportional version of RCV can disarm gerrymandering and produce fairer representation for racial minorities, even with race-blind maps. They make gerrymandering near impossible while automatically achieving the goals of the Voting Rights Act without consciously drawing maps for minority representation. This was confirmed in the 2024 multi-winner Portland, OR, city council elections.
Mathematics also helps with democratic decisions not reducible to a single election. Citizens’ assemblies and other deliberative mini-publics bring together representative groups of citizens to learn about public issues and develop recommendations. More than a thousand such processes have taken place worldwide. The lottery that chooses the participants is itself a deep mathematical problem of randomly selecting a panel that mirrors the public while different groups volunteer at different rates. New algorithms give every volunteer the fairest possible chance while hitting demographic targets, and now run real assemblies around the world through free, open-source software.
Even artificial intelligence, so often cast as democracy's next great threat, looks different through the mathematics lens. Democratic life happens largely in language — citizens explaining, objecting, proposing — and AI makes it possible to listen at scale. Mathematics is developing formal models and explicit guarantees that keep AI a translator between human expression and democratic decision, its workings open to inspection rather than hidden in a black box.
The founders believed self-government could rest on reason. Two and a half centuries later, that ideal remains as relevant as ever. Mathematics cannot replace politics or absolve us from the hard work of citizenship. But when nearly 80 percent of us worry about where democracy is headed, and its traditional guardians are absent or compromised, it matters enormously that rigorous, tested ideas for fairer systems exist.
Thinking mathematically is empowering. Math turns democracy from something that happens to us into something we can examine, question, and redesign. Once we understand how the machinery works, it loses its air of inevitability – and fixing democracy starts to look possible.
Ismar Volić is a Professor of Mathematics and the Director of Institute for Mathematics and Democracy at Wellesley College.
