Revised June 3, 2018; January 19, 2020
1. What is the largest number you can think of? Well, just add one, and you have a larger number! There are some large numbers that we are aware of. With record budget deficits of the developed nations in trillions of dollars (and the total nominal value of all the derivatives traded around the world approaching a thousand trillion dollars, or a quadrillion dollars), trillion and quadrillions are indeed large numbers.
- Even though a trillion rolls off the tongue not very different from a billion, a trillion is much bigger than a billion. If you spend a billion dollars a day, it will take a thousand days to spend a trillion dollars.
How to Represent Large Numbers?
2. The number of molecules in a cubic centimeter of gas is 2.7 x 1018 or 2.7 x 10^18 (this a simple way to express big numbers; instead of writing 10000, we write 104 or 10^4). Since electrons are even smaller, you would think there would be a humongous number of electrons in the observable universe; the estimated number is around 1087. That is, of course, a huge number, but is not infinity by any means.
- That should give you an idea of the power of an exponent. Each time the exponent goes up by one, the number becomes ten times bigger. So, even though 1087 may not look that big compared to 1018, it is a humongous increase. Another large number should be the distance from the Earth to the edge of the observable universe, and it is estimated to be about 46 billion light-years or around 1023 miles. Even though such large numbers are hard to be contemplated in our minds, they are all finite.
What Is a Googol?
3. There are some famous large numbers. A Googol is 10100, which is unimaginably vast compared to even the number of electrons in the universe (1087). As an aside, the internet company Google was to be named Googol, but someone made a mistake, and Google was the name given. A Googolplex is a whopper; it is 10Googol or 10(10^100). There are many such “famous large numbers.”
- Yet, you can add one to any of these large numbers and always get a bigger number. Therefore, no number, however large, is still finite.
Infinity – Beyond Any Conceivable Number
4. So, the mathematicians coined the term “infinity” to denote an indefinitely high number; The word comes from the Latin infinitas or “unboundedness.” Since infinity is uncountable, it has some strange characteristics: whatever you add to (or multiply by) an infinity (even if it is another infinity), you still end up with infinity.
- The famous German mathematician David Hilbert illustrated the “abnormal” properties associated with infinity using the idea of an “infinity hotel,” which has an infinite number of rooms. The “infinity hotel” always has a vacancy: the management can always ask the person occupying the Nth room to move to the (N+1)th room, (N+1)th room to move to the (N+2)th room, and so on, and thus give the Nth room to the new guest. Even if an infinite number of new guests arrive, the hotel can accommodate all of them!
Infinity Is Real
5. That is not to say that infinity is a useless or bogus concept. The arguments described above are valid. Mathematicians cannot do many integrations without infinity. Physicists use infinity all the time (but they try to end up with finite physical values).
- The concept of infinity is real (and weird). For example, a line of any finite length has an infinite number of points, whether it is an inch or thousand miles. The invention of calculus by Newton and Leibniz helped to handle some of the problems arising from such situations.
Space and Time – Infinite!
6. In the physical sense, infinity is a rather vague concept meaning “larger than anything that could in principle be encompassed by experience.” For example, space is infinite, and as far as our sophisticated instruments allow us to “see,” there is no end.
- Our universe is possibly infinite in extent since the scientists can “see” only to a finite extent. So, space is unlimited.
- What about the time? If our universe started at the Big Bang, that inflationary theory says there are multiple, parallel universes.
- According to the “cyclic theory” model, which is an alternate theory, the same universe comes to a “Big Crunch,” which leads to another Big Bang, and the whole process keeps repeating. So, there is no beginning to time either; time is infinite.
- By the way, both those theories are not correct, according to Buddha Dhamma. It is individual star systems (called “Cakkāvāta“) that undergo the birth-destruction cyclic process.
- I will write more in the future, but see the discussion: “Multiverse: Different Physical Laws and Different Dhamma?“. It is the lifetime of a Cakkāvāta (like our Solar system) that is called a Maha Kappa (great eon) in Buddha Dhamma.
Samsāra (Rebirth Process) Goes Back Infinite Time!
7. The Buddha used a great eon as the measurement unit to help his followers visualize the enormous length of samsāra. A great eon (maha kalpa or maha kappa) is said by the Buddha to be longer than the time it would take a man to wear away a mountain of solid granite one yojana (about 7 miles) around and one yojana high, by stroking it once every hundred years with a silk cloth.
- These days scientists use the word “eon” to denote the duration of a universe (from the “big bang” either to a “big crunch” or just fading away). But a Maha Kappa means the lifetime of our Solar system. Our universe has “no beginning.” In the future, it will be shown that the “Big Bang Theory” is not correct (which says that our universe came to existence from nowhere in a “Big Bang”.) See the discussion: “Multiverse: Different Physical Laws and Different Dhamma?“.
Buddha’s Analogy Consistent With Modern Science
8. Just for fun, I estimated the mass of the material that needs to be removed by the silk cloth each time (this happens every 100 years). Using a 7-mile cube of stone with a density of 2515 kg per cubic meter, I calculate the mass of the mountain to be 3.5 x 10 ^6 kg.
- Assuming the lifetime of our Solar system to be 5 billion years, I calculate the mass removed by each stroke is about 36 grams or about 1.2 ounces. That appears to be a reasonable number!
- When we try to visualize the wearing of a mountain, we can imagine how long a time that is. Yet, that is still nothing compared to the length of the samsara. As I said, infinity is a concept that is hard to wrap one’s mind around!
Another Analogy of the Buddha
9. One day the Bhikkhus asked the Buddha how many “great eons” had already passed and gone by. The Buddha told them, “Suppose, Bhikkhus, there were four disciples here each with a lifespan of hundred years, and each day they were each to recollect a hundred thousand great eons. There would still be great eons not yet recollected by them when those four disciples pass away at the end of a hundred years. Because, Bhikkhus, this samsāra is without discoverable beginning”.
- A fascinating book that talks about such hard to grasp ideas (in science) involving infinity is “The Beginning of Infinity: Explanations That Transform the World” by David Deutsch.
- Infinity is a mind-boggling concept. For example, one question that comes up frequently is the following. We have had infinite time to attain Nibbāna. So, why have all living beings not attained Nibbāna yet?”
- I have discussed that at, “The Infinity Problem in Buddhism.”
10. Each of us (including all living beings today) has “lived” for an infinite time. There is no traceable beginning to life, per Buddha. See, “Origin of Life – There is No Traceable Origin.”
- During much of that time, each of us suffered much more than any brief stretches of pleasure. That is because births in “good realms” (like human, Deva, and Brahma realms) is rare. See, “How the Buddha Described the Chance of Rebirth in the Human Realm.”
- Most of our past births had been in the four lower realms (including the animal realm) where suffering dominates.
- That is why the Buddha admonished us to strive hard to attain Nibbāna to avoid such harsh suffering in future rebirths.
Detailed discussion at the subsection, “Origin of Life.”