Positional Value and Linguistic Recursion

Computation and Natural Language The confluence of linguistic and mathematical thought in ancient India pro-vides a unique view of how modern mathematics and computation rely on linguistic and cognitive skills. The linchpin of the analysis is the use of posi-tional notation as a counting method for ancient and modern arithmetical procedures. Positional notation is a primary contribution from India to the development of modern mathematics, and in ancient India bridges mathe-matics to Indian linguistics. Pān: ini’s grammar, while not thought of as math-ematical, uses techniques essential to modern logic and the theory of computation, and is the most thorough-going historical example of algorith-mic and formal methods until the nineteenth century. Taken together, modern logic and ancient algorithmics show how computation of all kinds is con-structed from language pattern and use. To set the stage we start with the contemporary idea that all kinds of mathematics can be thought of as sets of formulas or sentences expressed in some formal language. Such sets are called theories, and are often thought of as being algorithmically generated by some precise rules of proof, such as the rules of predicate logic applied to domain-specific, or ‘‘non-logical,’ ’ axioms with specially defined terms. So there are theories of arithmetic based on axioms for addition and multiplication; set theories based on axioms for set membership and formation; theories of the real numbers; various kinds of geometry, algebra, and so on. Today such proof systems can also be thought of as computations, which mainly means spelling out the details by which a