Jain Ganit Ki Apratim Dharaye

Here is a comprehensive summary in English of the provided Jain text:

The book "Jain Ganit ki Apratim Dharaye" by Lakshmichandra discusses the development of mathematics in Jain literature through two primary streams: Pure Mathematics, referred to as "Rashi Siddhanta" (Theory of Quantities/Sets), and Applied Mathematics, known as "System Siddhanta" (Theory of Systems).

The author contrasts the modern development of Pure Mathematics, specifically the study of infinity by Georg Cantor (1845-1918), with the Jain approach. Cantor dedicated his life to establishing the theory of infinity, facing considerable challenges. Today, this developed theory of quantities forms the fundamental basis of all sciences, arts, and commerce.

The text highlights that Jain philosophy also dealt with infinities through its Rashi Siddhanta, employing concepts of analogy (Upama) and numerical proofs (Sankhya Praman). Works like "Trilok Pragnyapti" and "Trilok Saar" exemplify this endeavor. This Rashi Siddhanta became the foundation for the Karma Siddhanta (Theory of Karma).

The author posits that mathematics can determine the "dynamics" of any system. Applying Rashi Siddhanta to the Jain Karma System is presented as an unprecedented and even underdeveloped art and science in the world today. The abstract approach, utilizing modern mathematical concepts, is how mathematical system theories are bound through Rashi Siddhanta.

The text then delves into the "mysterious" aspects of this mathematics, which are difficult to translate into language. These mysteries lie in the mathematics of Yoga (union/practice) and Moha (attachment/delusion), their control, and the mathematics of automation. The dynamics of karma—its coming, bondage, manifestation (uday), and shedding (nirjara)—are mathematically modeled based on eight fundamental karmic natures. Jain philosophy has addressed the most difficult questions concerning the state, associated regions, and the mathematical aspects of karma's "anubhaga" (intensity or consequence) through mathematical and intangible means, focusing on the fundamental manifestations of atoms and the soul within dynamic karmic bonds.

The ultimate goal was to unravel the complexities of the impermanent and visible, leading to the revelation of the permanent and true. While all sciences have historically struggled with accounting for Utpad (creation) and Vyay (destruction), they have failed to account for Dhrouvya (permanence). Jain philosophy's mathematical science achieved this unique and unparalleled feat by providing a mathematical framework for Dhrouvya.

The mathematics of the Karma Siddhanta is presented as applied mathematics, forming the blueprint for modern system theory, which has been developed over the last 30 years. This applied mathematics primarily concerns control, achievement (labdhi), and automated instruments, all of which are rooted in the developed applied form of Rashi Siddhanta.

In conclusion, to comprehend the fundamental and profoundly mysterious depth of Jain philosophy, two efforts are evident within Karananuyog (the section of Jain scripture dealing with actions and their consequences):

1. The origination and application of analogies and numerical proofs.
2. The measurement of both visible and invisible forms of karma and their varied applications.

The measures of "dravya" (substance) were bound and depicted in the "map of the Trilok" (three worlds) through measures of substance, space, time, and disposition (dravya, kshetra, kal, and bhava). Here, algebra and geometry brought forth the mathematical types of the invisible and visible through the forms of both infinities and finite quantities. This was not the most difficult task; the greater challenge was to bind the dynamic aspect of "karma" (action) into Rashi Siddhanta. However, grasping Dhrouvya (permanence) through wisdom is presented as natural.