Philosophical questions about minds and computation need to focus squarely on the mathematical theory of Turing machines (TM's). Surrogate TM's such as computers or formal systems lack abilities that make Turing machines promising candidates for possessors of minds. Computers are only universal Turing machines (UTM's)—a conspicuous but unrepresentative subclass of TM. Formal systems are only static TM's, which do not receive inputs from external sources. The theory of TM computation clearly exposes the failings of two prominent critiques, Searle's Chinese room (1980) and arguments from Gödel's Incompleteness theorems (e.g., Lucas, 1961; Penrose, 1989), both of which fall short of addressing the complete TM model. Both UTM-computers and formal systems provide an unsound basis for debate. In particular, their special natures easily foster the misconception that computation entails intrinsically meaningless symbol manipulation. This common view is incorrect with respect to full-fledged TM's, which can process inputs non-formally, i.e., in a subjective and dynamically evolving fashion. To avoid a distorted understanding of the theory of computation, philosophical judgments and discussions should be grounded firmly upon the complete Turing machine model, the proper model for real computers.