Jain Mathematical - Book Summary

Here's a comprehensive summary of the provided Jain text, "The Jaina Mathematical Philosophy" by Prof. L. C. Jain:

The article "The Jaina Mathematical Philosophy" by Prof. L. C. Jain explores the sophisticated mathematical and philosophical underpinnings of Jainism, drawing parallels between ancient Jain thought and modern mathematical concepts, particularly in the realm of set theory and systems theory.

1. Introduction: Bridging Mathematics and Philosophy

The author begins by referencing Bertrand Russell's view of mathematics as a form of symbolic logic, highlighting the philosophical dimension of mathematics.
Jainism is presented as a philosophical tradition that developed a set-theoretic system theory, notably its Karma Theory. Texts like Ṣa¢khņdāgama, Mabābandha, and Kasāyapābuda (compiled around the beginning of the Christian era) are cited as foundational.
The Jain school developed post-universal measures through "existential sets" (cardinal numbers) and "constructive sets" (ordinal numbers), detailed as upamā māna and samkhyā māna in Karņānuyoga texts like Tiloyapaņņatti (5th century AD) and Trilokasāra (11th century AD).
The development of these ideas involved extensive commentaries, with Dhavalā and Jayadhavalā (9th century AD) being significant. Later, mathematicians like Nemicandra Siddhänta Cakravartı (11th century AD) formalized these concepts in works like Gommagasära and Labdhisāra.
Key pioneers in systematizing this mathematical philosophy include Mādhavacandra Traividya and Cãmundarāi (11th century AD), building on the works of Kundakunda (3rd century AD), Samantabhadra, Yativīşabha (5th century AD), and Akalarika (8th century AD). Keśava varņi (14th century AD) consolidated these into Karnātavșttisi, and Ṭodaramala (18th century AD) further refined the mathematical philosophy.

2. Modern Context: Paradoxes and Jain Solutions

The modern era faced paradoxes and antinomies in set theory, initiated by Georg Cantor, which led to debates about the nature of infinity.
Jainism, in contrast, had already developed a set theory that extensively used mathematically comparable infinities through "monads" or "units" representing space, time, phases, matter, and motion. These are termed pradeśa (point), samaya (instant), and avibhāgi praticcheda (indivisible corresponding sections).
The Jain approach aimed to avoid contradictions by establishing principles for mathematical operations on these infinities.
Logic, particularly the principle of syādvāda (the doctrine of manifold predication), played a crucial role in eliminating inconsistencies and contradictions.
The concept of omniscience in Jainism is viewed as a supreme, adaptable set of indivisible sections representing all knowledge, applicable to the Karma Theory.

3. Set Theoretic Approach

The article draws parallels between Jain mathematical concepts and modern set theory, noting that while systems like Russell's Principia Mathematica were found to be incomplete (as per Gödel), Jain set theory has been applied even to religious philosophy.
The fundamental Jain term for "set" is rāśi, with synonyms like samüba, ogha, puñja, etc.
Jain texts classify sets into various types: unitary elements, fundamental measure units, fixed fluent sets, point sets, instant sets, smallest/biggest/intermediary sets, pull sets, indivisible-corresponding section sets, transfinite sets, Karmic sets (like varga, vargaņā), and variable sets.
Jain analytical methods include reductio ad absurdum, one-to-one correspondence for transfinite sets, and methods of measure (pramāņa), reason (karaņa), explanation (oirukti), abstraction (vikalpa), cut (Khaņdita), division (bhājita), spread (viralana), and removal (apabsta).
Comparability of sets (alpabahutva) is a key concept, detailing relationships like "small," "equal," "smallest," "infinite times," etc. This concept is applied to cardinal and ordinal aspects of sets.
Topological sequences (Dharās) are discussed, with mention of "dyadic sequences" leading to the set of omniscience.
Jain principles offer explanations for paradoxes like Zeno's, by postulating finite points within finite segments, though acknowledging that infinite sets can be represented analytically. The statement about future instants being infinitely more numerous than past instants is presented as a postulated existent.

4. Systems Theoretic Approach

The Jain Karma Theory is compared to modern system theory, focusing on inputs, outputs, and state transitions. The Karma theory is described as a self-reproductive system, a concept modern science is still developing abstract models for.
The Karma theory involves operators (like Yoga and Moha), quantitative norms, measures, and various states and transitions.
Key elements of the Karma theory include:
- Yoga (volition) and Moha (charm) as operators with quantitative norms.
- A tetrad of measures: configuration (praksti), points (pradeśas), life-time (sthiti), and energy levels (anubhāga-ansa).
- Causality of simultaneous events linked to a behavioral time scale.
- The dual system of bios and karmic-matter phase-rise acting as feedback.
- Time-lag before karmic display, state-existence (sattva), and output values.
- Impedance (samvara) as an input function.
- Various measures (fluent, quarter, time, phase) of bios, non-bios, their merits/demerits, influx, impedance, decay, bond, and emergence, forming statistical surveys of karmic universes.
- Operational phases of bond, including bonding, state transition, rise, subsidence, etc.
- A sequence for the annihilation of the tetrad of bonds.
- Annihilation of state, rules for life-time cut, down-tract, and reduction of recoil energy.
- Three key operators (adhabpravṛtta, apurva, anivrtti) leading to correct vision.
- Complete emergence leading to omniscience.

5. The Mathematio-Logical Development

The Syādvāda system of predication was crucial in expressing the Karma Theory before the development of symbolic norms.
Jain texts describe various types of knowledge and modes of expression, leading to propositionality and logic in symbolic form.
The concept of dravya (substance) and paryāya (mode/modification) is explained, highlighting how a single entity can be considered "one" from the perspective of substance and "many" from the perspective of its various modifications and qualities.
Virasena's explanation of the fineness between space and time, distinguishing between "quarter-measure" and "fluent-measure" based on the number of ultimate particles, is discussed.
The article notes that P.C. Mahalanobis found relevance between Syādvāda and probability, suggesting the fourth Syādvāda category (inexpressibility/indeterminacy) provides a logical foundation for probability. However, it questions how causality fits into this.
J.B.S. Haldane also linked Syādvāda to the quantitative aspect of indeterminate solutions in equations, suggesting its interest to scientists for conclusions intermediate between certainty and uncertainty.
The article concludes by emphasizing the need for a team to explore the mathematical aspects of Jain philosophy, particularly its revelation in the Karma Theory, which also laid the groundwork for astronomical and cosmological systems.

In essence, Prof. L. C. Jain's article argues that Jainism possessed a highly developed mathematical philosophy, particularly evident in its Karma Theory. This philosophy engaged with concepts akin to modern set theory, transfinite numbers, and systems theory, offering sophisticated frameworks for understanding reality and avoiding logical paradoxes through principles like Syādvāda. The article posits that these ancient Jain mathematical ideas hold significant relevance for contemporary mathematical and philosophical inquiry.