In a somewhat polemical article on ‘Solstices, Equinoxes, and the Presocratics’ D. R. Dicks has recently challenged the usual view that the Presocratics in general, and the Milesians in particular, made significant contributions to the development of scientific astronomy in Greece. According to Dicks, mathematical astronomy begins with the work of Meton and Euctemon about 430 B.C. What passes for astronomy in the earlier period ‘was still in the pre-scientific stage’ of ‘rough-and-ready observations, unsystematically recorded and imperfectly understood, of practical (...) men’ whose chief concern was to fix the seasons for ploughing, seed-time, sailing voyages and religious festivals. Ionian speculation, says Dicks, took very little note of such observation: ‘some of its wilder flights of fancy might have been avoided, if it had taken more’. In this account of the rise of Greek astronomy, the natural philosophers have no part to play. Their theories represent a speculative enterprise without a scientific future, a philosophic sideline with no impact on the development of observational science from Hesiod to Meton or the development of mathematical astronomy from Meton to Ptolemy.I believe that such a dichotomy between early philosophy and early science in Greece is misguided in principle, and that it seriously distorts our picture of the initial phases of each discipline. It also imposes a considerable strain upon our credulity. Take the case of Anaxagoras who, according to Plato and Theophrastus, had given a causal explanation of eclipses of the moon a generation before Meton. (shrink)

Pāṇini's fourth century BC Sanskrit grammar uses rewrite rules utilizing an explicit formal language defined through a semi-formal metalanguage. The grammar is generative, meaning that it is capable of expressing a potential infinity of well-formed Sanskrit sentences starting from a finite symbolic inventory. The grammar's operational rules involve extensive use of auxiliary markers, in the form of Sanskrit phonemes, to control grammatical derivations. Pāṇini's rules often utilize a generic context-sensitive format to identify terms used in replacement, modification or deletion operations. (...) The context-sensitive rule format is itself defined using Pāṇini's more general method of auxiliary markers, the latter used to define many dozens of linguistic categories and rules controlling derivations of Sanskrit sentences through the manipulation of ‘non-terminal’ and ‘terminal’ symbols. This technique for controlling formal derivations was rediscovered by Emil Post in the 1920s and later shown by him to be ca... (shrink)

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 Pāṇini’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 Pāṇini’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. (shrink)