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Since introduction of the IEEE 754 floating point standard in 1985, numerical methods have become ubiquitous --- and increasingly sophisticated. With growing code complexity of numerical libraries grows the need for rigorous Software Engineering methodology: as provided by Computer Science and state of the art regarding digital processing of discrete data, but lacking in the continuous realm. We apply, adapt, and extend the classical concepts --- specification, algorithmics, analysis, complexity, verification --- from discrete bit strings, integers, graphs etc. to real numbers, converging sequences, smooth/integrable functions, bounded operators, and compact subsets: A new paradigm formalizes mathematical structures as continuous Abstract Data Types with rigorous Turing-computable semantics but without the hassle of actual Turing machines.