The Original Biggest Numbers - Numberphile

This discussion explores the history of contemplating very large numbers, moving beyond the recent discoveries in mathematical logic, such as Graham's number and Rayo's number, which date from the mid-20th century. The focus shifts to ancient civilizations, specifically highlighting the contributions of India, and within India, the Jain religion, which dates back two and a half thousand years BCE. Jainism, still practiced by millions, developed a rich tradition of mysticism that included incredibly large numbers, often used to represent vast periods of time. One such concept is the "paleopama," or "pit year," a unit of time derived from a thought experiment. It begins with a cubic pit, one yojana wide. A yojana is defined as slightly more than 10 kilometers, so for calculation purposes, it's rounded down to 10 kilometers. This pit, 10 km by 10 km by 10 km, is filled entirely with lamb's wool. The rule is that once every century, a single strand of lamb's wool is removed. The pit year is the total time it takes to empty the entire pit. To grasp the scale of this, a calculation is performed, making a significant underestimate for the number of strands. Assuming each strand occupies a cubic millimeter, and acknowledging the immense pressure at the bottom of a 10 km deep pit would compress the wool significantly, this assumption still allows for a calculation. A 10 km cube contains (10,000 meters * 1,000 millimeters/meter) cubed cubic millimeters. This results in 10^18 cubic millimeters, or 10^18 strands. Since one strand is removed every 100 years, the total time is 10^18 * 100 years, which equals 10^20 years. The speaker initially states 10^23 years, which would imply a slightly different initial volume or removal rate, but the magnitude remains immense. This "pit year" is presented as an absolute minimum of 10^23 years. These immense time periods were not merely mathematical exercises but were integral to Jain religious mythology. They were considered real periods of time, used to date the universe since creation. Given the vastness of the universe's timeline in Jain cosmology, even the paleopama proved insufficient, leading to even larger units. The next unit of time is the "suro prama," or "ocean year," which is simply defined as 100 million paleopamas. This means it's at least 100 million * 10^23 years, or 10^31 years. Jain mythology posits that the universe operates on a cycle that began approximately a quadrillion (10^15) ocean years before the present. This translates to roughly 10^15 * 10^31, or 10^46 years, marking their equivalent of a "Big Bang," though they conceived of it as an endlessly repeating cycle. Beyond these units, the Jains considered periods of time that encompassed more than an entire cycle. One such unit is the "pervanga," defined as 84 * 100,000 "pervies." A "pervy" is a fundamental unit of time, equivalent to 756 * 10^11 days. The term "100,000" is a "lakh," so it's 84 lakh pervies. The next level of magnitude is obtained by squaring, leading to "one perver," which is (8,400,000)^2 pervies. This process of increasing the exponent continues, culminating in the "sherapalika," or "top riddle," which is (8,400,000)^28 * 756 * 10^11 days. This "top riddle" translates to approximately 10^206 years, a period so vast it would extend far beyond the projected evaporation of all supermassive black holes in the universe. The "pervanga" itself, the lifespan of the original founder of Jainism, is over a quintillion years. The discussion then moves to numbers contemplated for their own sake, not just as measures of time. The ancient Jains developed a theory of "uninnumerable numbers," which are finite but so large they are practically infinite. This concept differs from modern mathematics. A description of the "first uninnumerable number" comes from Nemichandra's book "Trilocasara" (Essence of Three Worlds), written around 1000 CE. This number is conceptualized through an elaborate thought experiment involving a mystical geography. The central point is Jamboo Island, 100,000 yojanas wide (over half a million miles). Surrounding it are alternating oceans and islands, each double the width of the preceding one, creating exponential growth in size. The experiment begins by digging a cylindrical pit under Jamboo Island, 1,000 yojanas deep (5,000 miles), and filling it with mustard seeds. The height of the mustard seed mountain in the pit is 1/11th of the pit's circumference, making it thousands of miles tall. This initial mountain already contains an enormous number of seeds, enough to fit planet Earth multiple times over. The clever part involves distributing these seeds. The first seed is placed on Jamboo Island, the second in the first ocean, the third on the next island, and so on, until the entire mountain of seeds is exhausted. This process leads to a new, extremely wide disc (island or ocean), which could be 10^10^40 light-years wide, dwarfing the observable universe (10^11 light-years wide). Under this new disc, another pit of the same depth is dug, and a new mountain of mustard seeds is created with a height of 1/11th of its circumference. This second mountain is astronomically larger than the first. The process of distributing seeds ring by ring is repeated, leading to yet another location and another mountain. This entire process is repeated a colossal number of times: the cube of the number of mustard seeds in the original mountain. Given that the first mountain contains roughly 10^45 seeds, the process is repeated (10^45)^3 times, or 10^135 times. The final number generated by this entire procedure is the "first uninnumerable number," called "jaga parita asamata" in Sanskrit. Mathematician Radha Char Gupta, in 1992, analyzed this number. It is so large that it cannot be written using traditional mathematical notation. It requires Knuth's up-arrow notation, approximated as 10 ↑↑ (10^135), where the height of the exponential tower is 10^135, and each value in the tower is 10 ↑↑ (10^45). While incredibly vast, this number, and others from ancient Jain mathematics, are still significantly smaller than modern "big numbers" like Graham's number. However, they are large enough to challenge traditional notation and represent a crucial step in the development of conceptualizing immense quantities. The Jains were unique in their era for exploring such scales, with some accounts suggesting they even reached the equivalent of three Knuth arrows (10 ↑↑↑ 38), two levels beyond exponentiation. This intellectual leap by the ancient Jains predates similar explorations in the rest of the world by thousands of years, with others only catching up in the latter half of the 20th century.