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		§3.9 Second Paradigm: Identification of Functions

		The theoretically possible realities proper to our second paradigm are total recursive functions, introduced in Chapter 2. Many of the paradigms defined later in the book bear likewise on functions, so it is appropriate to begin our discussion by considering their intended interpretation.

		§3.9.1 Recursive Functions as Possible Realities

		Scientists often investigate physical systems that implement functions on a suitably chosen domain. The volume of a gas, for example, varies functionally with its temperature and pressure, and the electrical response of a photoreceptor depends functionally on the incident light. Such functions can be approximated with arbitrary precision using rational numbers, which may in turn be coded as natural numbers. In this way, scientific inquiry may often be represented as an attempt to discover which numerical function is actually implemented by Nature. To simplify our discussion we assume that the functions in question are total recursive mappings from N to N. Recall that the class of all such functions is denoted by . As with the earlier limitation to recursively enumerable sets (see Section 3.1.1 above), restricting attention to represents the nontrivial assumption that many natural phenomena can be construed as computational processes. Such will be our supposition.

		The class may also be considered from the point of view of automatic program synthesis. In this case, the role of Nature is played by a person P who has knowledge of some functional relation but cannot easily write a program to compute f (perhaps for lack of expertise, or simply time). We assume that P's knowledge of f amounts to P being able to respond to any argument with the value f(n). We seek a computerized system that discovers a program for f on the basis of the pairs fed to it. Once again we assume that P. is ''mechanical" in an appropriate sense, so that P's knowledge of f guarantees f's computability. (For more discussion of program synthesis in the context of the theory of empirical discovery, see Shapiro [172] and Jantke [98].)

		There is also a linguistic interpretation of . Some theories of language take a sentence to be an ordered pair of representations, one superficial, the other located at a deeper linguistic level. The latter representation stands in close relation to the meaning of the sentence, the former to its phonological realization. It is further assumed that the relation between underlying and superficial representations is a species of functional dependence, different languages implementing different functions of this kind. Competence