by Suzanne Zhang, Class of 2017

As a girl who has been in touch with Buddhism for her whole life, I am not unfamiliar with the two Buddhist vocabulary: Bodhisattva and Arhat. They are both two very common words in Buddhism. Since I was young, my father has been telling me that I have to be just as compassionate as a Bodhisattva and I need to have the self-discipline just like what an Arhat has. Moreover, I have always imagined that an Arhat would become a Bodhisattva before he reaches the stage of a Buddha. Unfortunately, my concept was wrong.

Recently, during Ms. Tran’s Pre-Calculus class, I have only learned that Arhats would become Buddhas directly without going through the stage of being Bodhisattvas. Therefore, technically, Bodhisattvas and Arhats are somewhat of the same level since they are both a stage before becoming Buddhas. However, if we understand it this way, why do they have different names then? And what exactly are the differences between the two different stages of sages? I believe we can find out the reasons by making comparisons between the two functions that the Pre-Calculus students must be familiar with, the exponential function( f(x)=e^x ) and the logistic function( f(x)=1/1+e^-x).

Exponential functions can be often observed in the real world. For example, statisticians always use the exponential function to model populations. As for logistic functions, they are the best model for pediatricians to model the growth of babies. Now, before going into depth, I want to do the simplest analysis and comparison. Both the exponential function and the logistic function only exist in the first and the second quadrant, which means that the range, which is also the outcomes are all-positive. To my own understanding, everything that Bodhisattvas and Arhats do are all positive. They do the things that benefit the living beings, all the righteous things. This is the first reason that I match the two functions with the two stages of sages.

By thinking of the graphs and looking at the shapes of the two functions, my mind-set has paired the exponential function with Bodhisattvas and the logistic function with the Arhats automatically. The reason for this is that by looking at the graph of an exponential function, we can see that the right side of the exponential function extends to infinity, which is just like how the Bodhisattvas always have infinite compassion and kindness towards all living beings. Most of Bodhisattvas have made the vow that before all the living beings are being crossed over, they will not become Buddhas. From this vow, we can see how great they are. They are willing to sacrifice many things for us. At this point, the Arhats only think of crossing over themselves. However, I do not take this as a symbol of selfishness, it is just that they believe when they cannot even cross themselves over, then whom else can they save? Even though this is not considered selfishness, compare to what the Bodhisattvas are like, they are still not wise enough. Just like the end behavior of the logistic function: as x approaches positive infinity, f(x) does not reach positive infinity like what the exponential function does but one. Thus, Arhats’ powers of crossing over living beings are limited.

What does limited mean? Limited means that it’s bounded – both above and below. Therefore, the logistic function has two horizontal asymptotes. One is y=o, the lower bound and the other is y=1, which is also the upper bound of the function. Whereas for the exponential function, it has only one horizontal asymptote, which is like what I have mentioned at the beginning, it’s because everything that they do has a positive outcome. However, the upper bond does not only symbolizes the limit to Arhats’ ability to cross over the living being, but also how they are bounded by the way of their cultivation; they are bounded by all the precepts. Arhats follow their precepts very well. Most of them cultivate the ascetic practices. Since I was a young girl, my father has told me to learn how to self-discipline like the Arhats do. Relatively, although Bodhisattvas do have precepts but they actually know what they should not do and what they are doing. Thus, they are not bounded by the precepts.

The asymptotes lead us to the next topic, the end behaviors of the two functions. We can take the negative infinity of the x-value as the past and the positive infinity of the x-value as the future. As x approaches negative infinity, which goes back in time when both the Bodhisattvas and the Arhats have not yet started their cultivation, their accomplishments were close to zero. Nevertheless, as time passes by, their cultivation goes into depth and their understanding of life gets more and more profound, both of the sages would earn nirvana and end the cycle of birth and death. Now, the difference comes in. As x approaches the positive infinity, the y-value of the exponential function would also reach the positive infinity; whereas the y values would just keep on approaching one. I take the y-value as the numbers of living beings they will cross over. By the time the Bodhisattvas enter nirvana, infinite lives would be crossed over. However, Arhats would only have themselves saved. Therefore, the other name for Arhats is the self-enlightened one (direct translation from Chinese). In the small vehicle, the Arhats are considered to be the highest level that one can achieve. However, when it comes to later where the great vehicle entered the world, people then only realized that the ones who have the abilities to help others are greater than the ones who only have the abilities to help themselves.

Now, let us move on and let us go to the most basis things of functions – the domain and the range.  I have already explained about the range and the domain. But, the other information that we can get from the two things is the functions’ continuity. The two functions are both continuous through all the domains. After contemplating for a while, I have come to a small enlightenment: if I say the x-values represents the time, then that means Bodhisattvas and Arhats never stopped their practice. Just like everything else, if we do things on-and-off-ly, then we are not going to get any better at that subject. We have to learn what the Bodhisattvas and Arhats do: they never hold their precepts for three days and stop for two days; they never stop crossing over living beings. We have to be consistent on everything. This applies to Mathematics as well. We have to practice math problems before we start getting the hang of solving different problems.

After the continuity, it comes the increasing or decreasing behaviors. Both graphs of the functions are always increasing throughout all the domains, but they do in different ways. I take the increasing of y-values as the increasing of their merit and virtues. For Bodhisattvas, as their cultivations gets more advanced and the more living beings they save, their merit and virtue increases with an accelerating rate. On the other hand, the Arhats’ merit and virtues increase most effectively when they are getting enlightened. But the most important thing out of all, their merit and virtues will never decrease. It is not like someone’s money that can be stolen, others cannot take one’s merit and virtues. What one owns belongs to himself or herself only. You have what you have, you cannot deny it; and if you have the heart to gain more, there are many things that you can do to obtain more.

Another thing that the exponential function and the logistic function have in common is that both of them do not have a any extrema, neither the definite ones nor the local ones. Yes, this is just what the Bodhisattvas and the Arhats are like. In Buddhism, we never talk about extremities. No matter what, we go for the middle way. For example, if a string instrument has tight string, they would snap easily. Contrarily, if the strings are to lose, they do not make any music. For this reason, we should go the middle way. From another perspective, I believe that the situation of none extrema symbolizes that all the Bodhisattvas and Arhats are humble. All the extrema are very obvious, which can be comprehended as showing off. Therefore, this situation shows us their humility. To learn to become sages, we have to get rid of our extreme personalities and emotions first. Without the extrema, both graphs have the perfect curves that show us what are the qualities that they need in order to be Bodhisattvas and Arhats. The nice curves symbolize a mellow and easy-going personality. They also symbolize the word “圆满” which means the great perfection. Buddhas are the only ones who have reached the state of the great perfection. Bodhisattvas and Arhats are both enlightened sages, they have almost reached the great perfection, and they will reach the stage eventually.

Now, I have a question for you, so, are the two functions even or odd? According to the definitions of even and odd functions, they are neither. For a function to be an even function, it has to by symmetrical over the y-axis; and in order for a function to be and odd function, it has to pass the origin. Since both functions do not fulfill any of the requirements, they are nether even nor odd. I take the even and odd behavior as metaphors for genders in humans. We humans are separated into two groups, the males and the females. However, I do not consider the Bodhisattvas and the Arhats any of the two genders, because they can appear in front of us in any forms: any gender, any race/ethnicity, and any age. Different people need to be crossed over with a different strategy. This is the thing that belongs to one of the Universal Worthy Bodhisattva’s 10 Great Vows, the ninth vow, “to accord with living beings.”. Hence, the Bodhisattvas and the Arhats are defined by any genders because their genders cannot be fixed.

And the last but not the least, the symmetry of the two functions; we all know that the exponential function is not symmetrical and the logistic function is symmetrical at some point. What can we get from this characteristic? Symmetry means that the two sides are equivalent to  each other, which is just like standing in front of a mirror and looking at oneself. Only people who are self-conscious would like to look at themselves from the mirror. From my point of view, Arhats still have a good amount of self-consciousness. They are not as selfless as the Bodhisattvas. That is why they still have the concern of how can they save others before saving themselves. And this concern, this self-consciousness might just be the thing that bounds their abilities. Bodhisattvas never think of themselves first, they just try their best to save as many as living beings as their abilities allow them to.

I have already analyzed the two types of functions and their connections between the two types of sages, the Bodhisattvas and the Arhats. Now, it has come to the time where I need to clarify the connection between this essay and its title:

I chose the title – “the Extra Parts”— because the exponential function is part of the logistic function. Or we can see from the other way around, the exponential function has gotten rid of  many parts that it does not need, the extra parts. It is true that Bodhisattvas have less humanly characteristics than Arhats do. For example, Bodhisattvas are not bounded like Arhats are, they live to eternity; or they have ended the cycle of birth and death. The numerator of one in the logistic function, which the exponential function does not contain, gives that bound. The other thing is that in the denominator of the logistic function, e^-x has an extra one (1+). Now, let me break it down first. Since e is an irrational number, then e^-x would also be an irrational number. Thus, e^-x stands for a whole. However, when you add a one to it, then the one becomes something extra. That one can be representing a bad hobby or habit, and it can also represent an attachment that you cannot get rid of. This means that Arhats still have some unwholesome habits or attachments, which are the ones that the Bodhisattvas have already got rid of. In addition, the negative sign, which is placed in front of the x in the power of e, is also a sign of imperfection. No matter what, I believe that after some vigorous cultivation, the Arhats would eventually get rid of all the habits from the “Suo Po” (Saha, or Samsara) world and become Buddhas. This same rule also applies to us humans. We are just like the polynomial functions with the power of infinity; we have so many imperfections, but as long as we have the faith and the heart, we would eventually become enlightened.

As an ordinary person, I admire and respect all the Bodhisattvas and the Arhats. However, what amazed me the most is the Six Parameters – the practices of giving, upholding precepts, patience, vigor, Samadhi, and wisdom, which the Bodhisattvas practice and how Arhats gain their enlightenment through contemplating the Four Noble Truths. Just like the exponential function, when the x-value reaches the infinity, the y-value would also reach infinity; Bodhisattvas also practice infinitely of the six paramitas. While they are practicing these skills, they gain infinite merit and virtue, and they also cross over infinite living beings. At this point, we can clearly see that the effort of practicing the six paramitas is the domain, or the input; and the merit, virtue they get and the living beings that they save are the range, or the output. This comparison can be made between Arhats and the logistic function as well. Arhats contemplate a lot about the Four Noble Truths, and in the end, they themselves would eventually get enlightened. However, no matter how hard they practice, they themselves would still be the only ones who get enlightened. Thus, in the logistic functions, as the domain gets greater and greater, the output would only approach one.

From what I have learned through this comparison and analysis, I am determined to be an exponential function. I believe that one shall not be bounded by our environment, the people around us, and neither by ourselves. We shall do things that benefit all the living beings instead of only ourselves.