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In Chapter 1 we introduced the idea of a paradigm for scientific discovery and enumerated the concepts that must be specified before a paradigm takes on mathematical meaning. A set of tools for building the needed concepts was described in Chapter 2. In the present chapter we deploy some of this apparatus in the investigation of two fundamental paradigms and some variants thereof. They are called: |
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• identification of languages; and |
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• identification of functions. |
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These paradigms are the barest and simplest to be encountered in this book, yet their underlying structure will show up in all of our ensuing work. |
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Pursuant to the analysis in Section 1.2, the paradigms are presented by specifying their component concepts, namely: |
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3.1 (a) a theoretically possible reality |
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(b) intelligible hypotheses |
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(c) the data available about any given reality, were it actual |
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(d) a scientist |
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(e) successful behavior by a scientist working in a given, possible reality |
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We shall devote considerable discussion to 3.1a-e in the context of language identification. Our discussion of function identification—and of subsequent paradigms in later chapters—can then be briefer, since many of the same considerations carry over to the new settings. The ideas discussed in this chapter derive from seminal papers by Gold [80] and Blum and Blum [18]. |
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The theoretically possible realities proper to our first paradigm are languages, in the technical sense of Chapter 2, so we begin with their intended interpretation within the paradigm of language identification. |
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§3.1 Languages as Theoretically Possible Realities |
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Languages can arise as objects of inquiry within either a scientific or a developmental context. We consider these two contexts in turn. |
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