|Abstract:||Is thought computation over ideas? Turing, and many cognitive scientists since, have assumed so, and formulated computational systems in which meaningful concepts are encoded by symbols which are the objects of computation. Cognition has been carved into parts, each a function defined over such symbols. This article reports on a research program aimed at computing these symbolic functions without computing over the symbols. Symbols are encoded as patterns of numerical activation over multiple abstract neurons, each neuron simultaneously contributing to the encoding of multiple symbols. Computation is carried out over the numerical activation values of such neurons, which individually have no conceptual meaning. This is massively parallel numerical computation operating within a continuous computational medium. The article presents an axiomatic framework for such a computational account of cognition, including a number of formal results. Within the framework, a class of recursive symbolic functions can be computed; formal languages defined by symbolic rewrite rules can also be specified, the sub-symbolic computations producing symbolic outputs which simultaneously display central properties of both facets of human language: universal symbolic grammatical competence and statistical, imperfect performance.