Review of Henry Thomas Colebrooke's 1817 Translation of Sanscrit Works on Division by Zero

2026-04-16T07:23:30+00:00

We review the introduction of the number zero and of total arithmetical operations of addition, subtraction, multiplication, and division in various Sanscrit works from 628 CE to 1621 CE, translated into English in the book "ALGEBRA WITH ARITHMETIC AND MENSURATION, FROM THE SANSCRIT OF BRAHMEGUPTA AND BHASCARA" by Henry Thomas Colebrooke, 1817 CE. We note the existence of intermediate steps in the development of zero from a placeholder into a number. We introduce the concepts of a paraconsistent arithmetic that is uniquely determined but which contradicts its axioms, and of a paralogical arithmetic that is uniquely determined but which is non-logical. We find that Brahmegupta, 628 CE, described two total paraconsistent arithmetics of fractions that contain the lexical operations of real arithmetic. Bhascara, 1150 CE, used only one of Brahmegupta's fractional arithmetics, but we generously credit Bhascara with making this arithmetic consistent. This manoeuvrer makes both of Brahmegupta's total fractional arithmetics consistent. However, Bhascara also introduced an inchoate arithmetic of infinitesimal tuples, which reintroduced Brahmegupta's inconsistency. We suggest that deterministic arithmetics are useful, despite any logical shortcomings. We illustrate this with the IEEE 754 Standard for Floating-Point Arithmetic, which is useful because it succeeds in its aim of specifying a deterministic arithmetic, despite the fact that this arithmetic is non-logical. We conclude that Brahmegupta and Bhascara produced a more logically sound total arithmetic than the IEEE standards committee. We recommend that computer arithmetic is founded on transreal arithmetic, which surpasses the Sanscrit arithmetics in its application to calculus and mathematical physics.

The Accusation Threshold Problem for Accusational Utterances

2026-07-31T13:30:26+00:00

Having previously defined accusations, we noticed that accusations appear in gradations leading to borderline cases. The resolution of boundaries between accusations and other utterances, which do not qualify as accusations, is a critical matter for some applications of accusation theory.

We take ``accusational utterances'' as the widest class of utterances that purport to be accusations. Accusations are a subclass of accusational utterances.
Agents may differ in their willingness to
recognzize an accusational utterance as an accusation. An agent absorbs an accusation if they recognize the accusation as such and do not disqualify it as a mere accusational utterance.

In some cases it is hard to understand whether or not an accusational utterance will be absorbed by an agent. The accusational threshold problem for an accusational utterance concerns the question under which conditions an accusational utterance qualifies as an accusation and thereby is worthy of absorption.

An accusation may or may not be valid, where validity concerns the truth of the body of the accusation.
Rejecting, or merely doubting, the validity of an accusation is a proof of absorption of the accusation. Admitting the validity of an accusation is a sign of absorption as well.

Within the class of accustions we find a strict subclass of verdictional utterances, which are accusations for which the accuser is not in significant doubt regarding the validity of the accusation. The verdictional utterance threshold problem is about the demarcation between accusations and verdictional utterances.

Transmathematica

Review of Henry Thomas Colebrooke's 1817 Translation of Sanscrit Works on Division by Zero

The Accusation Threshold Problem for Accusational Utterances