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.
2. The number of molecules in a cubic centimeter of gas is 2.7 x 1018 (this a simple way to express big numbers; instead of writing 10000, we write 104).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. This is of course a very large number, but is not infinity by any means.
- This should give you an idea of the power of an exponent. Each time the exponent goes up by one, the number becomes 10 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.
3. There are some famous large numbers. A Googol is 10100, which is unimaginably large 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 that was 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.
4. So, the mathematicians coined the term “infinity” to denote an indefinitely great 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 a “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. In fact, even if an infinite number of new guests arrive, the hotel can accommodate all of them!
5. This is not to say that infinity is a useless or bogus concept. The arguments described above are totally 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 in length or thousand miles in length. Invention of calculus by Newton and Leibniz helped handling some of the problems arising from such situations.
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. Besides there are possibly infinite numbers of universes as well. So, the space is infinite.
- What about 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 whole process keeps repeating. So, there is no beginning to time either; time is infinite.
7. The Buddha used a great aeon as the measurement unit to help his followers visualize the enormous length of sansara. The length of a great aeon (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 “aeon” to denote the duration of a universe (from the “big bang” either to a “big crunch” or just fading away).
- 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 universe to be 30 billion years, I calculate the mass removed by each stroke is about 12 grams or about 0.4 ounces. This appears to be a reasonable number! So, a kalpa in Buddhism turns out to be approximately an aeon as perceived by the scientists. When we try to visualize the wearing off a mountain we can imagine how long a time period 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!
8. One day the Bhikkhus asked the Buddha how many great aeons 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 aeons. There would still be great aeons not yet recollected by them when those four disciples pass away at the end of hundred years. Because, Bhikkhus, this sansara is without discoverable beginning”.
- An interesting 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; see, “The Infinity Problem in Buddhism“.
9. Here is a bit longer video on why it is not possible to discover either spatial boundaries of our universe or find a “beginning” to time, because universes come into existence all the time.