Mathematics is the lingua franca of the sciences. Few speak it. Today’s statesmen and the handful of courtiers they have time to trust must often go beyond their expertise. This is the Courtier’s Conundrum: how can the inexpert adviser advise expertly? Margaret Thatcher’s six policy advisers were not scientists. Yet they often gave scientific advice, because they had to. They applied what they knew to what they did not. Second-guessing the specialists to whom ministers will otherwise defer requires a consciously “back-of-the-envelope” and yet objective deployment of mathematics – a simple but not naïve quantitative inspection of policy options. Checking the tire-pressures is better than merely kicking the tires. On the other hand, neither statesmen nor their advisers have time or competence to reinvent the wheel. Simple calculations done honourably and properly can prevent large and wasteful errors. For example, the cost of abating CO2-driven warming turns out greatly to exceed that of focused adaptation to the damage the warming may cause. Mathematical simplification of complex issues is no panacea and can be abused, but some attempt at rigour is preferable to the merely qualitative, partisan approach that is customary. Uncosted ideology is costly. It is immoral too. More information on this lecture is available on the University of Western Ontario event webpage.