The analysis of problematic mathematical texts, particularly from India, has required the introduction of a new category of rigorous discourse, apodictic discourse. We briefly recall why this introduction was necessary. We then show that this form of discourse is widespread among scholars, even in contemporary Mathematics, in India and elsewhere. It is in India a natural outgrowth of the emphasis on non-written communication, combined with the need for freedom of thought. New results in this first part include the following: (i) {Ā}ryabha___a proposed a geometric derivation of a basic algebraic identity; (ii) Brahmagupta proposed an original argument for the irrationality of quadratic surds on the basis of his results on the varga-prak___ti problem, thereby justifying his change in the definition of the word kara___{ī}.