Spiritual Buddhist Wisdom

Buddhism and Mathematics

by Cherry Chong, Class of 2016

Everything in the world is interconnected- humans and animals, animals and environment. In this universe, everything is related to each other. Just like what Buddhism called as cause and effect, karma, and retribution. And this reminds me of a Buddhist quote that says, “Everything in the universe comes from the same root.” At the beginning, I cannot understand what it meant. But as I grew up and started to gain more knowledge on Buddhism, I realized that it is actually true. This interconnection within the universe includes the relationship between Buddhism and Maths. Although one is a religion, another is a study, they are still deeply related to each other. One example that I will be stating here would the the relationship between the Bodhisattva and Arhat; exponential function and logistic function. The Bodhisattva represents the exponential function while the Arhat represents the logistic function.

The exponential function can be defined as the form f(x)=a*b^x, where “a” does not equal 0, b>0, and b does not equal 1, and they are all real numbers. “b” is the base of the exponential function. Thus, the Bodhisattva is the base of the universe. The Bodhisattva is compassionate and tried to save all living beings. The name “Bodhisattva” can be translated into “enlightened being”. It has two meanings. First, one who causes all sentient beings to become enlightened. Second, an enlightened one among all sentient beings. The Bodhisattva is a sentient being. But he is a sentient being who is enlightened. He is a cultivator among those who understand. And for cultivators, he is the one who truly practices. He “leaps out of the dust”, the dust is defilement. He cultivated to the point where the dust gets thinner till he can leap out of it. In the exponential graph, the graph begins at the bottom and then gradually increases until it reaches a point where it increases tremendously to positive infinity. That is the part where the Bodhisattva “leaps out of the dust”. And once he get out of the dust, he does not have to bother about food, clothing, and shelter. He can concentrate on cultivating the Six Perfections, which are giving, holding precepts, patience, vigor, concentration, and wisdom. And because there is not only one Bodhisattva in the universe, it cannot equal to 1, yet greater than 0. The Bodhisattva has only one attachment, which is to save all living beings in the Saha World. They are willing to guide the people through their stupidity and delusion. They tell people to practice the Six Perfections and Ten Thousand Practices daily. And with these two main points, which we can imagine as the x- and y- axis, the graph manages to ascend quickly towards positive infinity. Without the x- and y-axis, it is impossible to even form a graph. Similarly, it is important to practice the six perfections and ten thousand practices daily without fail, or else it will be impossible to achieve the goal of enlightenment. And to practice it life after life- towards positive infinity- until they reached Buddhahood. And the Bodhisattva also teaches the people to bow to the Buddhas daily. Bowing can create limitless merit and virtues. Therefore, bowing is the input of the exponential function, “x”, and once you put that number into the function, the “f(x)” will be a lot greater than the original input. So, “f(x)” is the output, or also the merit and virtues we get from bowing. The Bodhisattva believed in the Four Noble Truths, impermanence, suffering, selflessness, and Middle Way. They practices the Buddhist Path to liberation, perfections, mindfulness, wisdom, and compassion.

The Four Noble Truths are The Truth of Suffering, The Truth or Accumulation, The Truth of Extinction, and The Truth of the Way. In The Truth of Suffering, it is at the beginning of the graph, where it never touches 0. Because there is never a beginning. The reason why there is not a beginning is that the beginning is the end and the end is the beginning. Without a the end, there will not be a beginning; without a beginning, there will not be an end. Thus, the beginning equals to the end. That is why the exponential function does not have a beginning nor an end. And after a long time suffering at the bottom, humans started to accumulate blessings and plant good roots. Then, they started to practice the way to enlightenment. And once it reaches a certain point, the world does not exist to them anymore- emptiness is form; form is emptiness. When that happens, they get Enlightened, which is the part of the graph where it increases to positive infinity. Thus, once a person attains the state of Bodhisattva or Buddha their lives will be limitless, and they will live a life of infinity. The graph of an exponential function looks like an single upward curve, where it never touches 0.

So, the number of living beings the Bodhisattva saves increases exponentially. He did what other people cannot do. That was what made his a Bodhisattva. When others find that helping ignorant beings to become enlightenment is difficult, a Bodhisattva is willing to take on that job. He patiently guides that ignorant being towards the right path. Thus it is not easy to be a Bodhisattva. He did not have a starting point, as the graph does not start from 0. That means that there was never a beginning for a Bodhisattva. As long as there is more than one living being in the world and more than one Bodhisattva in the universe, the Bodhisattva will keep on saving the living beings (a>0 and b>1 = exponential growth) He existed since the universe was created and saves uncountable beings since then. In f(x)=a*b^x, it means that living beings and Bodhisattva are the same in the universe- f(x). And together, they both multiply by the factor of x.

The logistic function can be defined as the form f(x)= c/(1+a*b^(x)), where a,b, and c are all positive constants, b<1, and c is the limit of the growth. As we can see, the denominator of this function contains the exponential function + 1. “Arhat” is a Sanskrit word with three meanings: worthy of offerings, without birth, and killer of thieves. Worthy of offerings mean that the arhat is worthy of accepting the offerings from people and gods. All he has to do is sit there and the people and gods will make offerings to him, because he is worth it. “without birth” means that he has already left the cycle of birth and death. So, he will not be born again. He was called the “killer of thieves” was that he killed the thieves of ignorance. However, what the arhat sees as thieves, the Bodhisattva does not see as thieves. The precepts of an Arhat is different from a Bodhisattva. In the opinion of the arhat, if he sees a thieve, he would kill him. But for the Bodhisattva, he will attempt to liberate the thieve. In other words, a arhat is act according the what he sees as true or false. They follows the precepts strictly. This is why in the logistic graph, there is a asymptote at the top of the line, which acts as a boundary for the line. This boundary is the limit of the arhat. In contrast, the Bodhisattva does not have a limit. He is limitless. He believes that he should use any possible methods to save the living beings, as long as the goal is achieved. That is why the exponential graph does not have an asymptote. Because of that idea, the Bodhisattva has the capacity to expand and increase the number of being liberated by him. While the arhats, bounded by his precepts and rules, cannot expand his boundaries in saving living beings, Speaking of the levels of Arhat, there are divided into four levels. If an arhat has attained the fourth fruition and does not continue to progress in his investigation, and did not advance in his cultivation, then he is called a “fixed-nature sound-hearer”. Basically, he was fixed on that level. This part reflects on the logistic graph, where the graph begins from the bottom and increases- the Arhat cultivates- until it reaches a certain level, it starts to stay constant- when the Arhat stays in the same level. Although his cultivation was not that advanced, he felt satisfied and did not consider making further progress. However, if he continues his cultivation, then he can attain the position of a Bodhisattva. For great arhats, they can be said to have attained the state of Bodhisattva. But they manifest as bhikshus  and do not practice the Bodhisattva way. And in their hearts, they have the magnanimity of a Bodhisattva. Thus, they can gradually attain the state of Bodhisattva. Actually, in their past lives they had already attained Buddhahood. But for the sake of propagating the dharma, they transformed into the body of a bhikshu to act as an Arhat and spread the Buddhadharma. Arhats do not have any outflows. He had already attained the state of patient with the non-production of Dharma. That reflects the asymptotes above and below. He does not have any more outflows, so the line of the graph does not exceed the asymptotes. In other words, an Arhat was more focused on his own cultivation instead of bothering to save many beings. But that does not mean that he does not save living beings. He does save living beings. But just not to the extend of Bodhisattva, who is totally committed in saving living beings. He is not busy. He is always free and totally at ease. He have not attachments, no cares, no self, no others, no living beings, no life, no nothing…… And because of this characteristic, the number of living beings saved by him was stunted-stays constant- instead of increasing exponentially to positive infinity. Saving living beings is not his job. His mind is always at ease. He does not have any false thinking. And when he enters samadhi, he can sit for thousands of years. This reflects the graph where the end behaviour is infinitely constant. That is his mind- calm and constant forever.

As we all know, the logistic graph looks like a horizontally stretched “s”.

It starts from the beginning without touching 0, and increases until it reaches a point, which is the horizontal asymptote, and stays constant. An Arhat is “not-born”. He do not get reborn again. Thus, there is no production and no extinction- negative infinity and positive infinity. There is no beginning for him. So, from the graph above, “a” is when he started cultivating as an Arhat and saving living beings in that process. Then between “a” and “b”, it is the part when his state of arhatship started to increase to his desired state. That desired state is defined as point “b”. That point is also where he stayed constant in saving living beings. That was because he was satisfied in his position currently and wanted to stay at that point. An Arhat starts without a beginning either. He starts to help living being in his process cultivation. And once he achieved his goal, the number of living beings he liberated would gradually stay constant. So, from that explanation, we can see that “c”(in the equation) is his limits.

In conclusion, the exponential function reflects the Bodhisattva. The Bodhisattva has a compassionate heart and had one attachment. And that attachment is to save all living beings. Most Bodhisattvas made vows that if one living being did not attain Buddhahood, then they will never attain Buddhahood. This shows that Bodhisattvas are devoted in saving living beings. Just like the exponential graph above. It had no beginning and no ending. Thus, a Bodhisattva does not have a beginning or end. They started from nowhere and ended at nowhere. As in the graph, the line never touches 0, which is its asymptote; and increases towards positive infinity. In other words, the number of living beings saved by the Bodhisattva is endless. It keeps increasing exponentially. As for an Arhat, he is not as devoted to save living beings. His focuses on attaining his goal. And once he reaches the state he wanted, he feels satisfied and stop attempting to cultivate to get to a higher level. This reflects the graph of a logistic function as a constant line at the end. He is carefree, compared to the Bodhisattva who tries to look for many different ways to save living beings. Therefore, the Bodhisattva is more vigorous in saving living beings, which led to the increase of liberated living beings to increase exponentially. In contrast, the Arhat is more focused on his own cultivation and has a carefree mind, which led to the constant positive end behaviour in the logistic graph with an asymptote as the limit. The one similarity between the Bodhisattva and Arhat is that both of them has already attained the Way.