An Indian Quasi-Fregean Theory of Number

Abstract

A very interesting account of the reference of number words in classical Indian philosophy was given by Maheśa Chandra (1836–1906) in his Brief Notes on the Modern Nyāya System of Philosophy and its Technical Terms (BN), a primer on Navya-Nyāya terminology and doctrines. Despite its English title, BN is a Sanskrit work. The section on “number” (saṃkhyā) provides an exposition of a theory of number which can account for both the adjectival and the substantival use of number words in Sanskrit. According to D. H. H. Ingalls (1916–1999), some ideas about the reference of number words in BN are close to the Frege–Russell theory of natural number. Ingalls’s comparison refers to a concept of number in Navya-Nyāya which is related to the things numbered via the so-called “circumtaining relation” (paryāpti). Although there is no theory of sets in Navya-Nyāya, Navya-Naiyāyikas do have a realist theory of properties (dharma) and their theory of number is a theory of properties as constituents of empirical reality, anchored to their system of ontological categories. As shown by George Bealer, properties can serve the same purpose as sets in the Frege–Russell theory of natural number. In the present paper, we attempt a formal reconstruction of Maheśa Chandra’s exposition of the Navya-Nyāya theory of number, which accounts for its affinity to George Bealer’s neo-Fregean analysis. As part of our appraisal of the momentousness and robustness of the “circumtaining” concept of number, we show that it can be cast into a precise recursive definition of natural number and we prove property versions of Peano’s axioms from this definition.