Mathmetical Philosophy In The Jaina School Of Thought

This document, "Mathematical Philosophy in the Jaina School of Thought" by L.C. Jain and S.K. Jain, explores the philosophical underpinnings of Jain mathematics, drawing parallels between its concepts and the development of Western mathematical philosophy, particularly set theory.

Key Arguments and Themes:

• Origin of Mathematical Philosophy: The authors link the term "Mathematical Philosophy" to Bertrand Russell, who saw it in the work of early Greek geometers as they moved from empirical rules to general propositions and axioms. They then posit that the Jaina School of Mathematics was also engaged in a similar philosophical pursuit, particularly within its "Karma theory" found in ancient texts like the Purvas.
• Jaina Approach to Logic and Language: The text highlights the Jaina concept of Syadvada (the doctrine of conditioned predication) as a parallel to the later Western philosophical understanding that deeper philosophical realms require symbolic representation to express propositions between truth and untruth. Syadvada, with its use of the word "syat" (meaning "perhaps" or "in some way"), allows for a nuanced, multi-ended perspective on an object's state, avoiding the "mono-ended pursuit" of older philosophies.
• Jaina Set Theory and the Innumerate/Infinite: A central theme is the Jaina engagement with the concepts of the innumerate (numbers that are too large to count but not infinite) and the infinite. The authors suggest that the Jaina School developed its own "set theory" (rasi siddhanta) to address these concepts within the context of their Karma theory. This theory deals with the perpetual cycle of births and deaths, and the annihilation of karma, requiring mathematical tools to categorize and calculate various sets of spiritual states and karmic influences.
• Addressing Cantor's Paradoxes: The document directly addresses the paradoxes encountered in Georg Cantor's set theory, such as Russell's paradox (the set of all sets that do not contain themselves). The authors argue that the Jaina approach, by dealing with "ultimate units" and constructing sets with "real existence" within their philosophical framework, inherently avoids these contradictions. For instance, they suggest that the set of "omnisciences of all the accomplished souls" in Jainism has only one value (omniscience itself), thus resolving the paradox of self-reference.
• Indivisible Units and Zeno's Paradoxes: The text draws a connection between the Jaina concept of indivisible units (like samaya - indivisible instant, and pradesa - indivisible space) and the attempts to resolve Zeno's paradoxes. While the Greeks struggled with the infinite divisibility of motion, the Jaina concept of indivisible units, fundamental to their understanding of reality and karma, offered a way to bypass these issues.
• Karma Theory as a Philosophical System: The Jaina Karma theory is presented not just as a religious doctrine but as a complex philosophical and mathematical system. The calculation of karma involves various measures and types of units needed to "annihilate the Karma state matrix." This, the authors argue, demonstrates the Jaina school's "own formalism of symbolism and its symbolic logic applied in the Karma theory," which, in Russell's terms, becomes mathematics.
• Jaina School's Positivistic Approach: The Jaina School is described as taking a "positivistic approach" in introducing the innumerate and infinite, aiming to explain "endless processes from ab aeterno to ad infinitum" and finding a mathematical path to "perpetual immortality."

In essence, the document argues that the Jaina tradition, through its intricate philosophical and mathematical systems like the Karma theory and its sophisticated handling of numerate, innumerate, and infinite quantities, was engaged in a form of "mathematical philosophy" comparable to, and in some ways anticipatory of, later Western developments in logic and set theory. The authors position Jaina mathematics as a distinct yet philosophically rich endeavor that offers unique solutions to problems that have challenged Western thinkers.