Not all infinities are made equal.
Now a mathematically well-known fact, this wasn’t a trivial claim to make back in 400 BCE, when the Jain mathematical text Surya Prajnapti was written. Accepted by both Shwetambars (Upang Agams) and Digambars (Karnanuyog), this Jain canonical text extensively defines mathematical principles that guide Jain cosmology.¹ While I have not read the text itself, the depth of exposition Jains have on mathematical and logical systems amazes me — it’s not something the average person (or, I, at least) would expect from a religion.
Amidst all its mathematical exposition, the Surya Prajnapti extensively defines large quantities. This is what backs much of the detail in Jain cosmology, especially for units of time and space.
In modern times, the concept of infinity is associated deeply with mathematics. However, philosophical experiments with unbounded quantities actually predate (and arguably motivated) subsequent mathematical experiments. Indians and Greeks were two of the earliest documented cultures to toy with the idea of infinity, and both did so from a philosophical angle, rather than a formalized mathematical one.
According to Jain mathematics, quantities can be divided into three major categories (each of which has several subcategories)²:
- Enumerable (Sankhya): quantities you can count to, like how many Dravyas (substances) there are in the universe
- Innumerable (Asankhya): practically infinite numbers that are rigidly bounded, mostly used to define smaller units of infinite quantities
- Infinite (Anant): boundless quantities, like how many souls there are
The bounds of each of these are defined in depth, mathematically. If you’re interested in learning more, check out this. For the sake of this article, I’ll focus on understanding them in context; after all, innumerable and infinite quantities already exist to represent numbers our brains cannot process.
Enumerable quantities are fairly straightforward to understand; for example, there are only six Dravyas (substances). But how much of each of those substances exist in the universe? Moreover, how big is the universe? Depending on your depth of Jain cosmological knowledge, you may know the answer to both of those questions is “infinite.” But these infinities are not all equivalent quantities.
What does it mean for there to be multiple infinities?
While space, time, matter, and souls are all infinite, they’re not all equal in quantity. While it is apparent that numerable quantities are smaller than innumerable ones, and these are all finite and can therefore be compared to each other, the same may not be obvious for infinities.
Take the graph below for example. Assume all the parabolas continue infinitely upwards. The blue is necessarily going to cover less space than the navy, which is less than the orange, which is less than the red.
Given that these parabolas are all in two dimensions, consider the tiers of infinity one can have in three or four dimensions. Exploding head emoji.
Asankhya vs Anant
Sankhya numbers are easy to understand; they’re countable. 1010 may be very large, but we know how to count to it in a finite amount of time. Asankhya and Anant are both infinite quantities, so what is the distinction?
In modern mathematics, numbers can be countable/uncountable and finite/infinite. The set of all natural numbers, for example, is countably infinite. The set of all real numbers, on the other hand, is uncountably infinite. (If we sat down to attempt counting the numbers between even 0 and 1, we could keep getting finer and finer forever.) And the set of natural numbers below 10, 100, or 1000 are all countably finite. A set cannot be uncountably finite.
There are three categories here, and three categories above, but they do not appear to correlate one-to-one. If Sankhya really constitutes all countable numbers, it would encompass both countably finite and countably infinite sets. Anant definitionally equates to infinite. What’s left for Asankhya, then?
Jain texts define the lower end of Asankhya numbers through exponents, and by modern mathematical definitions, are countable. The Asankhya subcategories range from very high countable numbers to transfinite numbers (infinite subsets or larger infinities).² According to Dr. Narendra Bhandari, however, Asankhya numbers are used to denote numbers that are indeterminable because of the constant changes, or Paryaya, occurring in their environment. He compares it to an inability to know the number of particles due to wave-particle duality and references Tattvartha-rajavartik as a source.³
While all of this doesn’t seem immediately relevant, I find that the extent to which Acharyas and Arihants have explored cosmology and mathematics fascinating; it elevates my respect for Jainism as a philosophy and contributor to modern scientific culture. The works of these Jain masters don’t exist in isolation from the rest of the world; much the opposite, they’ve played an active role in developing the foundations of modern-day mathematics. Moreover, understanding the expanse of time and space conveyed by these numbers puts into perspective just how long our souls have been cycling through lives, and how unique and rare of an opportunity a human life is to further our progress towards liberation.
¹Jain Study Center of North Carolina. (n.d.). Appendix — Summary of Swetambar Jain Agams. JainWorld.com. https://jainworld.com/literature/jain-agams/jain-agam-literature/upang-agams/
²U. (2011). Jainism, Key to Reality: Tattvārthasūtra by Āc Umā Swāmi. India: Digambar Jain Trilok Shodh Sansthan.
³Bhandari, N. (2015). Jainism: The Eternal and Universal Path to Enlightenment : a Scientific Synthesis. India: Prakrit Bharati Academy.
