The power of mathematics to cut across disciplinary boundaries seems almost unlimited. By some quirk of mathematical logic, the Schrödinger equation — with imaginary time — can offer an elegant description of an evolving biological population. Differential geometry has been fruitfully applied to finding the right price for financial products. Surprisingly, ideas from mathematical physics may even tell us something about the nature of democracy, and its inherent pitfalls.

In the eighteenth century, the French philosopher the Marquis de Condorcet proved a disconcerting theorem about public opinion. As individuals, people tend to be logically consistent in their opinions or preferences. If, in the current campaign for the US Presidency, I prefer Obama over Clinton, and Clinton over Romney, then I'll also prefer Obama over Romney. If A outranks B, and B outranks C, then A should outrank C. Most of our thinking conforms to this basic rule of logic.

But just because individuals follow such logic, this doesn't imply that groups do. Within a population of people, Condorcet proved, it is possible for a majority to prefer A over B, a majority to prefer B over C, and a majority also to prefer C over A — making a cycle of collective preference, so that it's impossible to say which the people really prefer. Indeed, they may well have no clear preference.

Condorcet's proof of principle doesn't establish that cycles really exist, but more recent analyses, based on statistical mechanics, suggest that it's likely. Suppose that individuals within a large population rank a series of alternatives, A, B, C... in random order, independently of one another. In this context, physicists Matteo Marsili and Giacomo Rafaelli asked, what is the chance, mathematically, that you'll find cycles in the group preferences when these people vote on the various alternatives in pairs? Their results showed that as the number of alternatives becomes large, the chance of having cycles rapidly approaches one (www.arxiv.org/abs/cond-mat/0403030).

There's a certain logic to the politicians' apparent love of inconsistency and waffle.

This suggests — within the limitations of the random-preference approximation — that the trouble presented by cycles may indeed be real. (One caveat: the work also suggests that the human tendency towards conformism may lead many people to similar preferences, thereby helping to avoid Condorcet cycles and to make the public's collective views more 'rational' than they would otherwise be.)

Even if work of this kind rarely makes specific testable predictions, it often offers new and surprising perspectives on old questions. For example, one might explore how a politician, uninhibited by the need for logical consistency, should behave if trying to appeal to a population with cyclic preferences. If the opinions of the majority involve some contradictions and apparently irrational cycles of preference, then a candidate trying to match the public as closely as possible might well have to abandon consistency and use more flexible and slippery tactics to appeal to as many voters as possible.

Some researchers have suggested that this might be related to our perpetual difficulties in working out precisely where politicians stand (www.arxiv.org/abs/cond-mat/9806359). Just maybe, there's a certain logic to the politicians' apparent love of inconsistency and waffle. They're merely responding to our own collective confusion.