The Infinity Problem in Buddhism

July 15, 2017; Revised February 5, 2018; Re-written March 23, 2021

We have been in the rebirth process for an infinite time. How is it possible that all of us have not attained Nibbāna? Here we will show that there is no contradiction.

Introduction – “Infinite Monkey Problem”

1. The question is: “The Buddha has said that there is no discernible beginning to the rebirth process (see the suttā in “Anamatagga Saṃyutta, Samyutta Nikāya). In other words, we have had an infinite time to attain Nibbāna. So, why have we all not attained Nibbāna yet?”

  • This issue has been discussed in discussion forums without reaching a conclusion. See, “The problem of infinity in Buddhism” at Dhamma Wheel and “The infinity problem in Buddhism” at the Sutta Central forum in 2017.
  • This question seems to have its origin in the “infinite monkey theorem,” which states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type any given text, such as the complete works of William Shakespeare.
  • By the way, this infinite monkey theorem is another evidence of how unimaginably large infinity is: “Infinity – How Big Is It?“.

2. First of all, the monkey theorem is based on a monkey generating random letters typing on a keyboard. It is assumed that the monkey will keep typing on the keyboard non-stop for an infinite time.

For example, the probability that the monkey will come up with the word “banana” would be less than 1 in 15 billion, but not zero. Thus it is a theoretically possible outcome, even though improbable.

  • But the enormity of it comes to view when you realize that you have to get a WHOLE TEXT correctly without making too many errors at a stretch (in one continuous period.) For example, to get TWO words correctly, like “yellow banana,” has a  probability that is the multiplication of the probabilities of getting each word right. The probability that a monkey getting those two words correctly is less than 1 in a billion-billion (1 in 10^18).  That is extremely small.
  • The above Wikipedia article says: “..the probability that monkeys filling the entire observable universe would type a single complete work, such as Shakespeare’s Hamlet, is so tiny that the chance of it occurring during a period of time hundreds of thousands of orders of magnitude longer than the age of the universe is extremely low (but technically not zero).”
  • That is because the age of our current universe is only about 14 billion years as estimated by science. Infinity cannot be assigned a number. Any large number you can think about, CANNOT BE the largest number because you can just add 1 to that number to make it bigger. There is no ending!
Two Relevant Issues

3. We will discuss TWO aspects of this issue.

  • FIRST, we will show that the MECHANISM of the infinite monkey theorem does apply to the present case.
  • SECONDLY, we will show that even if an infinite number of living-beings has attained Nibbāna, there will still be an infinite number left!
Attaining Nibbāna Cannot be Compared to a Random Event Like a Keystroke

4. Let us first discuss the differences between the infinite monkey theorem and the current problem. This will provide some insights to those who are not familiar with the actual process involved in attaining Nibbāna.

  • Attaining Nibbāna is NOT a process that can happen randomly. Typing a letter on a keyboard happens EACH TIME a monkey hits the keyboard. However, each life of a living-being CANNOT be taken as a “shot at Nibbāna.”
  • First of all, lives in the four lowest realms have ZERO chance of getting to Nibbāna. Even for lives at or above the human realm, a single ATTEMPT to attain Nibbāna is not possible without being explained the way to attain Nibbāna. That can be explained ONLY by a Buddha or a disciple of the Buddha who has attained at least the first stage of Nibbāna.
  • To put it another way: a monkey can hit the keyboard every time, i.e., it can ATTEMPT TO try to type. But a living-being WOULD NOT even know about Nibbāna in the absence of a Buddha. There could be billions of years at a stretch where even the word Nibbāna is not known. Thus, one could not even try to attain Nibbāna. It is like the monkey not being even able to hit a single key for billions of years at a stretch!
  • Therefore, attaining Nibbāna is a million-fold more difficult than a monkey being able to write Shakespeare’s Hamlet accidentally.

5. To even try to attain Nibbāna, first one must learn about Nibbāna from a Buddha or a true disciple of a Buddha. One must learn about the Four Noble Truths/Paṭicca Samuppāda/Tilakkhana AND THEN comprehend those.

There have been only 7 Buddhās within the past 91 mahā kappā. A single mahā kappa encompasses over 10 billion years.  This timeline is discussed in the “Mahapadana Sutta (DN 14)“.

  • There was only a single Buddha (Vipassi) in the mahā kappa 91 mahā kappās earlier. There was not a single Buddha in the next 60 mahā kappā,  and then two Buddhās (Siki and Vessabhu) appeared. Then there were 30 mahā kappā without a single Buddha, and so far in the current mahā kappa, there have been four Buddhās (Kakusanda, Konagama, Kassapa, Gotama). One more Buddha (Maithree) is expected to appear before this eon ends.
  • Even during a given Buddha Sāsana, only a fraction of HUMANS will get to hear/read about the true Buddha Dhamma. Today, the percentage of Buddhists worldwide is roughly 5%-10%, and much less than 1% have even heard about the true version of Buddha Dhamma.
Path to Nibbāna Is Not a Mechanical Process

6. Therefore, reaching Nibbāna CANNOT be considered a mechanical process and, thus, CANNOT be compared to a monkey hitting arbitrary keys on a keyboard to generate Shakespeare’s Hamlet.

  • We can consider a monkey typing a single letter to a living-being hearing the correct Buddha Dhamma during a lifetime after being born in a higher realm. Therefore, we can call that a “single-shot” at Nibbāna, corresponding to a single keystroke by a monkey.
  • However, a mathematician could still say that there will be an infinite number of such “single-shots” by a living-being over an INFINITE time.
  • In other words, even though two such “single-shots” may be separated by huge time spans, given infinite time, an infinite number of them can accumulate. Therefore, the premise of the infinite monkey theorem should still hold.
  • We went through the above discussion just to show that the two processes cannot be equated.
  • Now, we address the SECOND issue mentioned in #3 above.
Infinite Number May Have Attained Nibbāna

7. It is indeed possible that an infinite number of living beings HAVE ATTAINED Nibbāna in the past.

  • Not only that, an infinite number of living beings may have attained the Buddhahood in the past. Of course, attaining the Buddhahood is infinitely more difficult than attaining Arahanthood.
  • Therefore, the infinite set of living beings who have attained Nibbāna is “much larger” than the infinite set of living beings who have attained the Buddhahood.
  • The key to this puzzle is that there are “many levels of infinity.” It has been revealed by mathematicians within the past hundred years, thanks to the pioneering work of the mathematician George Cantor. See, “George Cantor – The Man Who Founded Set Theory.” The following video provides good insights too.

A Nice Visualization of Infinity Within Infinity – The Infinite Hotel Paradox

8. The following video explains why there can be types of “smaller infinities” within infinity. In particular, the set of positive integers is a “smaller infinity.” Those who have attained Nibbāna fall under that category. Regardless of how many have attained Nibbāna, more could attain Nibbāna.

  • The following video discusses an infinite number of buses filled with an infinite number of guests arriving at an infinite hotel. It is shown that the infinite hotel can accommodate all of them and more!

  • I have set the video to stop around 2:10 minutes. It is enough to see that the hotel can accommodate an infinite number of guests at any given time. During the presence of a Buddha Sāsana, only a finite number of living-beings (humans, Devas, and Brahmas) attain Nibbāna.
  • The rest of the video is more mathematical and shows that even an infinite number of buses with an infinite number of passengers in each bus can be accommodated! You can watch the whole video by clicking “watch on Youtube.”
Other Related Issues

9. Of course, several other questions now arise: Where do all these infinite numbers of living beings live? Do they all live in our Solar system? It will take many more future posts to explain these fully, but we can summarize as follows.

  • Brief answers to those questions are as follows: According to the Buddha, there are an uncountable number of planetary systems populated with living-beings. While an uncountable number of living-beings live in our Solar system, there are an uncountable number of such planetary systems (cakkavāla) in the world.  Each cluster of 10,000 such cakkavāla can have a Buddha appearing periodically. Thus, in principle, there could have been an infinite number of Buddhās.
  • Such details are in suttas in the Tipiṭaka, mostly in the Digha Nikāya. I have discussed one sutta briefly: “Buddhism and Evolution – Aggañña Sutta (DN 27).”
Additional Resource

Infinity is a very complex concept. There are many levels of infinity. Infinity minus infinity can still be infinity.

  • Therefore, even as there have been an infinite number of Buddhās and an even higher infinity of those who attained the Arahanthood, there are still an infinite number of living beings (including us) who have not yet attained Nibbāna. There is no contradiction.
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