Formal models of cultural evolution analyze how cognitive processes combine with social interaction to generate the distributions and dynamics of 'representations.' Recently, cognitive anthropologists have criticized such models. They make three points: mental representations are non-discrete, cultural transmission is highly inaccurate, and mental representations are not replicated, but rather are 'reconstructed' through an inferential process that is strongly affected by cognitive 'attractors.' They argue that it follows from these three claims that: 1) models that assume replication or replicators are inappropriate, 2) selective cultural learning cannot account for stable traditions, and 3) selective cultural learning cannot generate cumulative adaptation. Here we use three formal models to show that even if the premises of this critique are correct, the deductions that have been drawn from them are false. In the first model, we assume continuously varying representations under the influence of weak selective transmission and strong attractors. We show that if the attractors are sufficiently strong relative to selective forces, the continuous representation model reduces to the standard discrete-trait replicator model, and the weak selective component determines the final equilibrium of the system. In the second model, we assume inaccurate replication and discrete traits. We show that very low fidelity replication of representations at the individual level does not preclude accurate replication at the population level, and therefore, accurate individual-level replication of representations is not necessary for either cultural inertia or cumulative cultural adaptation. In the third model, we derive plausible conditions for cumulative adaptive evolution, assuming continuous cultural representations, incomplete transmission and substantial inferential transformations.