The Original Biggest Numbers - Numberphile

Key Concepts

• Jain Mathematics: An ancient Indian tradition that explored extremely large numbers, often categorized as "innumerable" (finite but practically infinite).
• Paleopama (Pit Year): A unit of time based on emptying a 10km³ pit of lamb's wool at a rate of one strand per century.
• Suro Prama (Ocean Year): A unit of time equal to 100 million Paleopamas.
• Innumerable Numbers: A classification for finite numbers so vast they exceed human comprehension or practical counting.
• Iterative Processes: The methodology of repeating an operation (like filling a pit) a number of times equal to the result of the previous operation, effectively creating exponential growth.
• Knuth’s Up-Arrow Notation: A mathematical notation used to represent extremely large numbers that exceed standard exponentiation.

1. Ancient Jain Time Scales

The Jain tradition utilized massive time units to describe the cyclical nature of the universe.

• Paleopama: By calculating a 10km x 10km x 10km pit filled with wool (assuming 1 strand per cubic millimeter), the duration is at least 10²³ years.
• Suro Prama: Defined as 100 million Paleopamas, totaling at least 10³¹ years.
• Cosmological Cycles: Jain mythology posits that the current cycle began approximately a quadrillion (10¹⁵) Ocean Years ago, placing the start of the cycle at roughly 10⁴⁶ years ago.
• Pervanga and Beyond: The largest unit, the Shera Pelica, involves squaring 8,400,000 repeatedly, resulting in a time scale of approximately 10²⁶ years. This exceeds the lifespan of supermassive black holes, describing a universe long after all matter has evaporated.

2. The First "Innumerable" Number

Mathematician and historian Radha Charan Gupta (1992) analyzed the Trilocasara (Essence of Three Worlds), written by Nemichandra around 1000 CE, which details the construction of the first "innumerable" number.

The Methodology:

1. Geography: The model uses a series of concentric islands and oceans, where each subsequent ring is double the width of the previous one.
2. The Pit: A pit 5,000 miles deep is dug under the first island (Jamboo Island) and filled with mustard seeds.
3. Distribution: The seeds are distributed one by one across the islands and oceans.
4. Iteration: Once the seeds are exhausted, a new pit is dug at the location reached, and the process repeats.
5. The Scale: The process is repeated a number of times equal to the cube of the seeds in the original mountain.

The Result: The final number is approximately 10↑↑(10¹³⁵) using Knuth’s up-arrow notation. This represents a tower of exponents so tall it cannot be drawn or written in standard notation.

3. Historical Context and Significance

• The Hiatus: The Jain tradition held the record for the largest numbers contemplated in human history for nearly two millennia. There was a significant "hiatus" where no other culture approached these scales until the mid-20th century with the advent of modern mathematical logic (e.g., Graham’s Number, Rayo’s Number).
• Mathematical Sophistication: While these numbers were rooted in religious mythology, the underlying logic—iterating arithmetical operations to create massive growth—is the foundational principle of modern large-number theory.
• Comparison: While modern numbers like Tree(3) or Graham’s Number are significantly larger than the Jain "innumerable" numbers, the Jain scholars were using similar iterative logic thousands of years before the development of modern notation.

4. Notable Quotes

• "The biggest numbers of the ancient world were in India... of all the very big numbers that got contemplated in India, the biggest come out of the tradition of the religion Jainism."
• "It is a testament to the size of those big Graham’s numbers that you did this crazy thing... and then you at the end said, 'Oh no, it’s still not close to those ones.'"

Synthesis

The Jain tradition represents a unique intersection of mysticism and advanced arithmetic. By conceptualizing time and space through iterative, exponential processes, they moved beyond simple counting into the realm of "innumerable" numbers. While modern mathematics has developed notation (like Knuth’s arrows) to describe even larger values, the Jain methodology of repeating operations to reach astronomical scales remains a foundational precursor to the study of large numbers in contemporary mathematics.