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The past decade has witnessed substantial progress toward the goal of constructing a machine capable of understanding colloquial discourse. Central to this progress has been the development and application of mathematical methods that permit modeling the speech signal as a complex code with several coexisting levels of structure. The most successful of these are "template matching," stochastic modeling, and probabilistic parsing. The manifestation of common themes such as dynamic programming and finite-state descriptions accentuates a superficial likeness amongst the methods which is often mistaken for the deeper similarity arising from their shared Bayesian foundation. In this paper, we outline the mathematical bases of these methods, invariant metrics, hidden Markov chains, and formal grammars, respectively. We then recount and briefly interpret the results of experiments in speech recognition to which the various methods were applied. Since these mathematical principles seem to bear little resemblance to traditional linguistic characterizations of speech, the success of the experiments is occasionally attributed, even by their authors, merely to excellent engineering. We conclude by speculating that, quite to the contrary, these methods actually constitute a powerful theory of speech that can be reconciled with and elucidate conventional linguistic theories while being used to build truly competent mechanical speech recognizers.