Paul Glimcher (2003) and Colin Camerer, George Loewenstein, and Drazen Prelec (2005) make powerful cases in favor of neuroeconomic research. Yet in their equally powerful defense of standard “Mindless Economics,” Faruk Gul and Wolfgang Pesendorfer (forthcoming) point to the profound language gap between the two contributing disciplines. For example, for an economist, risk aversion captures preferences among wealth lotteries. From the neuroscientific viewpoint, it is a broader concept related to fear responses and the amygdala. Furthermore, as economic models make no predictions concerning brain activity, neurological data can neither support nor refute these models. Rather than looking to connect such distinct abstractions, Gul and Pesendorfer (forthcoming) argue for explicit separation: “The requirement that economic theories simultaneously account for economic data and brain imaging data places an unreasonable burden on economic theories.” We share the conviction of Glimcher (2003) and Camerer, Loewenstein, and Prelec (2005) concerning the potential value of neuroeconomics, yet we believe that the field will live up to its potential only if a common conceptual language can be agreed upon. Hence, we face the Gul and Pesendorfer challenge head on by developing theories that simultaneously account for behavioral and brain imaging data. The principal innovation lies in our use of the decision theorists’ standard axiomatic methodology in this highly nonstandard setting. This removes any linguistic confusion by defining concepts directly in terms of their empirical counterparts. It also allows us to pinpoint how to design experiments directed to the central tenets of the theory, rather than to particular parametrizations. If these experimental tests reveal the theory to be wanting, then knowing which axiom is The Neuroeconomic Theory of Learning