Computability and logic: 3rd ed.
Computability and logic: 3rd ed.
The rediscovery of the mind
From the Church-Turing Thesis to the First-Order Algorithm Theorem
LICS '00 Proceedings of the 15th Annual IEEE Symposium on Logic in Computer Science
Computation: finite and infinite machines
Computation: finite and infinite machines
The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions
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This paper explains how mathematical computation can be constructed from weaker recursive patterns typical of natural languages. A thought experiment is used to describe the formalization of computational rules, or arithmetical axioms, using only orally-based natural language capabilities, and motivated by two accomplishments of ancient Indian mathematics and linguistics. One accomplishment is the expression of positional value using versified Sanskrit number words in addition to orthodox inscribed numerals. The second is P驴驴ini's invention, around the fifth century BCE, of a formal grammar for spoken Sanskrit, expressed in oral verse extending ordinary Sanskrit, and using recursive methods rediscovered in the twentieth century. The Sanskrit positional number compounds and P驴驴ini's formal system are construed as linguistic grammaticalizations relying on tacit cognitive models of symbolic form. The thought experiment shows that universal computation can be constructed from natural language structure and skills, and shows why intentional capabilities needed for language use play a role in computation across all media. The evolution of writing and positional number systems in Mesopotamia is used to transfer the thought experiment of "oral arithmetic" to inscribed computation. The thought experiment and historical evidence combine to show how and why mathematical computation is a cognitive technology extending generic symbolic skills associated with language structure, usage, and change.