Jaina Ulterior Motive Of Mathematical Philosophy

Here is a comprehensive summary of the provided Jain text, "Jaina Ulterior Motive of Mathematical Philosophy," by L.C. Jain and C.K. Jain:

The article argues that Jaina philosophy, particularly its theory of Karma, was deeply rooted in mathematical and logical reasoning, acting as a "science of sciences." This philosophical pursuit was not merely abstract but served a practical, "ulterior motive" of explaining and resolving complex phenomena through rigorous mathematical frameworks.

Key arguments and findings presented in the text include:

• Mathematics as Philosophy: The authors establish that mathematics, as a structured and connected whole, aligns with the philosophical pursuit of wisdom and truth.
• Jaina Philosophy and Mathematical Manoeuvres: The Jaina tradition, focused on the theory of Karma (action), systematically employed mathematical concepts and logic to understand natural phenomena, including the intricate workings of the soul and its interactions with matter.
• Evolution of Linguistic and Conceptual Universes: The text traces the development of Jaina thought through different conceptual "universes" or schemas used to describe reality. Initially, seven linguistic universes were used, but later thinkers like Nemicandra simplified this to two key schemas: niscaya (determinant) and vyavahāra (usage). The union of these "naya" (schemas) forms the universe of pramāņa (measure).
• Set Theory and System Theory Foundations: The Jaina approach is characterized as being set-theoretic and system-theoretic, incorporating logic. This is evident in their detailed enumeration and classification of "Rāsi" (sets) of various entities, including souls, non-souls, karmic particles, space-points, and time-instants. The text highlights that this predates modern set theory.
• Mathematical Concepts Employed: The Jaina texts extensively used concepts such as:
  • Cardinal and Ordinal Numbers: For measuring quantities of entities.
  • Infinities: Eleven types of infinities were defined and elaborated, with a particular focus on divergent sequences (fourteen in number) described in the Trilokasāra, which helped locate finite and transfinite sets. This is compared to Georg Cantor's work.
  • Comparability (Alpabahutva): A method for understanding the relative size of sets, analogous to modern syntopology. This involved detailed analysis of relationships like "small," "equal," "greatest," "least," and various temporal relationships.
  • Logical Methods: Including reductio-ad-absurdum, one-to-one correspondence (preceding Galileo and Cantor), measures, reasoning, abstraction, division, and spread.
  • Laws of Indices, Logarithms, and Continued Fractions: These were applied to finite and infinite bases.
• Addressing Paradoxes: The Jaina mathematical principles, particularly alpabahutva and sequences, provided frameworks to explain apparent paradoxes, such as those posed by Zeno. The concept of finite space-points and time-instants within finite segments was crucial.
• System-Theoretic Approach to Karma: The Karma theory itself is presented as a sophisticated system theory. Key elements include:
  • Operators: Yoga (action) and Moha (attachment) are identified as operators with specific norms.
  • System Components: Configurations, particles, life-time (sthiti), and energy-level (anubhāga) are treated as components.
  • Causality: The concept of simultaneity of events and causal relationships, where the "determinant" (cause) necessarily determines the "determinatum" (effect), is explored, allowing for both synchronous and sequential causality.
  • Input and Output Values/Functions: The theory describes how Yoga and Moha act as inputs that influence the system's state and outputs, with continuous feedback loops.
  • Operational Phases of Bonds: A detailed sequence of ten operational phases for karmic bonds is outlined, including bonding, state-transition, rise, subsidence, etc.
• Mathematical-Logical Development and Syādvāda: The Syādvāda (seven-fold predication) system is presented as a crucial logical tool. It provides a framework for expressing complex realities with multiple perspectives, avoiding contradictions. The authors link Syādvāda's concept of "non-assertorial" predication to modern ideas of probability and indeterminate solutions in mathematics, citing examples like the square root of minus one.
• "Artha Samdrsti" (Symbolic Norm): This term signifies the introduction of symbolic representations and norms rather than just symbolic logic, leading to a "post-universal mathematics."
• Philosophical Motivation: The "ulterior motive" is linked to the Jaina pursuit of Omniscience (Kevala Jnana). The comprehensive knowledge of all subsets of reality, including "indivisible-corresponding-sections," is seen as a driving force behind their mathematical investigations. This pursuit allowed them to boldly confront indeterminacy, paradoxes, and contradictions.
• Comparison to Western Mathematics: The text suggests that Jaina mathematics was remarkably advanced, with concepts like one-to-one correspondence for transfinite sets predating similar Western developments. The authors imply that the modern understanding of mathematics could benefit from a re-examination of Jaina foundational texts.

In essence, the article posits that the Jaina tradition developed a profound and systematic mathematical philosophy, deeply integrated with its spiritual and ethical doctrines, particularly the theory of Karma. This mathematical framework was not an end in itself but a tool to achieve a deeper understanding of the universe and the nature of existence, driven by the ultimate goal of liberation and omniscience.