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The Jaina School of mathematical philosophy, therefore, took a positivistic approach to introducing the innumerate and the infinite. They were meant to explain the endless processes from ab aeterno to ad infinitum, the relations between various sets involved in the realities of various life types. They had to find, mathematically, a path to perpetual immortality in which there was neither births, rebirths, or the agonies They were looking for a way to achieve the perpetual bliss, infinite power and total knowledge.
To do this, they created the indivisible system of units, as the indivisible instant (samaya) and the indivisible space (pradesa), this is somewhat similar to the problem of the Eleatic School's, and Zeno's, paradoxes. As most of us know Zeno presented a series of mathematical paradoxes that have baffled mathematicians and philosophers until Russell developed his theory of infinite regression ( innumerate regression). Prior to Russell both the Greeks, and those who followed, were obliged to leave most discourse on infinities to a simple statement of, "as small as we please and as great as we please." For example, we find the following -- which was considered to be a truism by Socrates rather than a series of paradoxes -- from Zeno (a student of Parmenides, 5th B.C.E.):
1. Dichotomy: There is no motion, for whatever is transformed into motion, it will be required to reach the middle (of the distance) before it reaches the end (and for reaching that half-way point it will have to reach half of the half-way point, and so on ad infinitum).
2. The Arrow: Zeno states, every object is either at rest or in motion when it occupies the space equal to its own. That object is in that space now (this instant) and always. The moving arrow, therefore, is at rest (and not moving). This is a paradox.
There are two more paradoxes of Zeno presented in the Jowett's translation of The Dialogues of Plato (Vol.2.) The two cases outlined above could not be explained away without the innumerate processes in the nature of motion of physical objects. Unfortunately, this did not allow for division ad infinitum. Such sequences which could have a finite sum may come under the sets with innumerate members. According to Socrates, however, Zeno's paradoxes were not directed against the Pythagorean Schools because they dealt with ultimate units. The Jaina School also dealt with Karma theory through the theory of ultimate units as we have seen already. Even the phases of the
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