045. Comparable

tldr: people aren’t a totally ordered set

1. There is a common structure across most areas of math– at least in the beginning. The commonality can be described as a two step process: 1) Define a new type of mathematical object, and 2) create relations between these objects. Wording it like this can feel vague, but this structure is nothing new. In high school algebra, we describe functions and relate them– adding, subtracting, defining equality; in college algebra, we define rings and create homomorphisms.

Consider the real numbers. First, we define what real numbers are (which we don’t *really* define until real analysis, but this isn’t the point). Then, we can try to relate them to one another– defining what it means to add, divide, multiply, and subtract. We also compare real numbers. What does it mean for 1<=2 (<= defined as less than or equal to)? Or 3<=100? Well, we can say a<=b if 0<=b-a. We are enforcing a way to define what it means for one real number to be bigger than another. 

Consider, yet another, more abstract, example. We define sets as a collection of objects. For our purposes, we can consider sets of numbers, such as A = {1,2,3} and B = {2,3,4}. Once we have these new objects, we want to know how they are related to each other. For example, the elements in common with both A and B are 2 and 3, and the only element in A but not B is 1. (We would describe these two relations as the intersection of A and B, and the difference A-B respectively.) In particular, we can impose a way to order sets, namely using subsets. For instance, we can define A to be a subset of B if every element in A is a subset of B (e.g., A = {1,2} =< B = {1,2,3}). Yet again, we have found a way to describe what sets are bigger than another.

What we have defined so far are partial orders for different objects. For the real numbers, we can always compare two elements– it is always the case that either a<=b or b<=a in the real numbers. Thus, we say that the real numbers are a totally ordered set. But this isn’t the case in sets under the subset partial order. Consider the sets A = {1,2,3} and B = {4,5,6}. Here, it is not the case that A is a subset of B, and it is similarly not the case that B is a subset of A– A and B are incomparable. And thus, sets are only partially ordered– not totally ordered.

2. I have always been a semi/extremely existential person (particularly more so when I was younger). I would spend evenings staring into a mirror and wondering to myself what it would feel like to not exist– picturing a vast and empty dark void in which my consciousness would spend the rest of eternity, only to come to the harsh realization that nothingness couldn’t ever properly be imagined. In some ways, this turned me into an optimistic person, believe it or not. To the best of my knowledge, this is the one life I was given, and I am going to enjoy it as much as I can. I chose the “Ignorance is Bliss” route because I want to be blissful. I wanted to be able to look in a mirror and not immediately question what is happening in the soul of a complex creature I don’t quite recognize.

But every then and again, I flashback to these long nights pondering life. The moments of existential dread have blended together into the mustard yellow of my childhood home’s bathroom. I don’t quite question life anymore. I am here, because if I wasn’t, there would be no “I” questioning why I am here in the first place. Instead, I am questioning who “I” is. I wonder why our actions in a finite lifetime should ever be perceived as important, and if said actions can be of finite value/worth, is this value valuable? Let’s start to address these questions one by one.

3. Who is “I”? Here, “I” is not I, but rather the word itself. Who does “I” describe? Certainly me, but then who does that describe? In 24.251, Philosophy of Language, we have been debating questions like this. In particular, we have been questioning the following two sentences:

  • Mark Twain is Mark Twain.
  • Mark Twain is Samuel Clemens

(For those who don’t know, Mark Twain was a pseudonym Samuel Clemens used.) The first sentence feels trivial. Of course, a=a— how could this *not* be the case? Yet, even though Samuel Clemens *is* Mark Twain, the second one loses this sense of triviality. And so, philosophers in philosophy of language have been arguing over what the difference is. Gottlob Frege had the idea that what makes these two sentences different depends on sense and reference. Sure, Mark Twain and Samuel Clemens may refer to the same person, but in some sense (including an emotional sense) they are different.

When I think of this topic, I keep coming across one question. Is the following sentence guaranteed to be logically consistent (i.e., does this sentence make logical sense?):

  • Mark Twain, the author, is happy, but Mark Twain, the father, is not.

I would argue that this sentence does make sense. But couldn’t one argue this is a case of the second sentence before?– How can it be the case that Mark Twain the writer is happy but Mark Twain the writer is not happy when they are the same person? Well, Frege could argue that these identities are fundamentally distinct; that they do not refer to the same person/object/being. But consider a similar question:

  • Clark Kent went into the phone booth, and Superman came out.

One can similarly ask if this sentence makes logical sense. Shouldn’t it be the case that either Clark Kent enters and leaves, or Superman enters and leaves, or, if not either of these two cases, that Superman and Clark Kent are fundamentally different entities?

Gottlob Frege may argue that they have the same reference but not the same sense. In other words, Frege may argue that when I refer to Clark Kent, I am actually referring to Superman, and vice versa, but what makes them different is their relationship to the world and the sense/emotions these identities evoke. (To be clear, we don’t quite know how Frege would respond to this sentence/similar ones as he never made his stance on this sort of question clear.) 

I, however, think it would be wrong to say Clark Kent and Superman both refer to the same person. I instead think that either Clark Kent or Superman does not refer to anyone.

Before I explain this stance, let me first point out that Frege did accept the possibility that there could exist objects/concepts with no reference. In fact, he embraced this belief. Just because there was no real life version/reference of Odysseus doesn’t mean that Odysseus is meaningless. On the contrary– the sense and emotions that arise from hearing the story of the Odyssey aren’t nothing. In the same way, Clark Kent and Superman both have *some* sense/emotion affiliated with them. Clark Kent reflects the human side of Superman– literally. Clark Kent is the version of Superman that allows him to falter and care too deeply about mortal humans. In the same way however, Superman is the version of Clark Kent who can fight for his morals and beliefs to defend the humans. Individually, these two identities aren’t void of meaning or connotation. But I still think that either Superman or Clark Kent don’t refer to someone (not in the sense that neither actually exist in real life, but rather that in the world of Superman one of them has no reference).

Instead, I think one is just an extension of the other. That either Superman is a role played by Clark Kent, or that Clark Kent is a disguise for Superman. To see my thought process, I implore the reader to consider another sentence:

  • I was born a boy, and I am now a woman.

Are both subjects distinct creatures/objects? Some transgender people do have this (totally valid) mindset. They were one person, and now they are someone else. Different identities, different references, both of which exist in their own respect. I, however, do not take on this view. My body has (as far as I know) been a continuous shift over my entire lifetime– and I don’t think there was this 5 minute interlude in which I changed in a telephone booth to transform from one identity to the other. More accurately, I would say that my identity as a boy never existed/had any reference. There was no child descended from my parents on June 20th who was a boy. This would be the analogous belief that Clark Kent doesn’t exist. Clark Kent has always been Superman, whether or not he is acting like it outwardly.

Now let’s turn back to existentialism. Suppose that I am right (as I am, as myself, inclined to do). In my present state, it feels right and natural to say that I was and always have been a girl/woman. I don’t plan on this ever changing. But now consider another aspect of myself which I hold to be true: I was and always have been interested in teaching. This is true– education has always played a big role in my life and I do want to one day be a teacher. But if in ten years I am not interested in teaching, what does that mean? Does that mean previous versions of myself (i.e. the current present version of me) doesn’t want to teach? She just believes she wants to? I don’t know. I have been so wrong about my identity in the past, that I worry I will prove myself wrong again in the future.

So who is “I”? I am the culmination of who I was since birth til now– compounding together into a weird mash up of an identity crisis and an optimist. The parts of my identity I hold to be true about myself have always been true– I have always been a girl, and I have always been interested in teaching. But who “I” is is always in flux. Perhaps it is naïve to ever wonder who I am for a second in the sense that who I am that second is different from who I am the second afterwards. Maybe my identity isn’t comparable to the previous versions of myself, or maybe my identity isn’t slightly comparable to who I will become. Perhaps, comparison between these two distinct identities is futile.

4. What, if anything, is worthwhile? When it comes to philosophy, the biggest complaint I hear from people is that the subject gives you more questions than answers. This is especially difficult in classes like philosophy in which you can hear two or more really compelling arguments to believe contradictory facts. To be completely honest though, I feel like I am left with more questions than answers in non-philosophy classes. Let’s, for instance, consider math (of course). Having done math research (not that I know what that actually means), I have become a lot better at trying to figure out the next “thing”. Trying to take what we have learned and take it a step further.

“What if here, the 2 was a 5?” “What if instead of a continuous function, we considered smooth function?” “Why doesn’t this weird thing happen?”

Most of the time, we never talk about these problems in class. It would be cool to take a class in which every question a student is interested in they could pursue. While I am sure some such classes exist, this isn’t quite my point. My point is: it is significantly harder to think about the math questions willy nilly– especially the questions that haven’t been answered. This isn’t the issue in philosophy. People complain that philosophy classes don’t give you answers but this is incorrect. They give you too many answers. Like a multiple choice question in which every option seems correct, but there is no d) all of the above.

Coming into MIT, I thought that trying to understand all of these options in a philosophical question was worthwhile. If I could, I would spend a lifetime asking “If a God exists are they moral?”– and I could read every paper and article on the topic until I come to a carefully selected opinion: Yes, No, or Uncertain. But I don’t have a lifetime to find the answers for myself. So is considering the arguments worthwhile? Well, I am uncertain, but I say Yes and No.

On the one hand, I don’t think anyone should simply choose to ignore other viewpoints. But on the other hand, I have been figuring out what arguments I don’t want to have with others, and what arguments I especially don’t want to have with myself.

I used to question everything I did. Why date now when in four years you’ll be going to grad school? Why should I go on a hike this Saturday when I have a PSET due on Sunday? More recently these questions have evolved. Are my actions impactful? What, if anything, is worthwhile? These new questions are the type that I can feel comfortable in taking a stance and ignoring other viewpoints.

  • My actions have an impact on the world. I shape and develop the world around me. I could (and have) spend a long amount of time wondering if everything is destined from the get go, or I can choose to be ignorant and move forward feeling Important. Debating my worth and my value or lack thereof only hurts me in the process.
  • Happiness is worthwhile. Not in the sense that I can ignore everything that makes me sad. But more so– it isn’t wrong to want to spend time with friends when I could be PSETting. It isn’t wrong to hope for the best. It isn’t wrong to prioritize my self-care before my schoolwork.

5. Recently, I have been wondering if there is any point to trying. It often feels like there will always be people better than me. Trying to get better at math. Trying to care about improving education and equity. Trying to care about others and myself. But, here’s how I have been thinking about it. Pick any quality of any person (if it’s quantitative even better). We can compare people (objects) based on this quality they have/don’t have. 

Consider the quality of academics. Academia wise, there will always be a person smarter than I am. I believe that this is a fact. We can impose a total order on people of this one quality (whether or not we ourselves have the ability to accurately quantify this quality is another question in itself).

But trying to compare entire people? Entire beings of different mindsets and life challenges and personalities cannot be compared. People do not form a totally ordered set. I can aspire to have the happiness of someone else. I can want to care more deeply. But I only need to “improve” in such ways if I want to. If I don’t want to, why bother comparing in the first place.

6. I keep comparing myself to other people. People comparing themselves to others is prevalent throughout society– it’s how people figure out where in society they want to fit. At MIT though, the comparison culture feels especially toxic. I find myself looking at other students with insane schedules that they are staying on top of, wishing I could do what they can, but struggling with my own work. But this just makes me feel bad. Like, I am always just a bit behind the curve. For this reason, I have been questioning if comparison is even worthwhile.

But then, I think about the people I really truly look up to. The people I strive to be more like every single day. When I think about these inspirational people, comparison feels optimistic, not toxic and self-deprecating. Throughout my life, I have struggled to figure out what should motivate me. For a long time academia was my sole motivator. But now that I am in college, my journey in academia is coming to an end. And I am so tired of comparing myself to others in school. So I have been trying to figure out what will motivate me in life.Two years ago, one of the kindest people I have ever known passed away– Dr. Adnan Sabuwala. Another September has come and gone. I feel like I am still dreaming– like I will find out that he was just on a vacation, and any day now he will be around the corner. What I found most inspirational about Dr. Sabuwala, was how deeply he cared. I never got to ask him why he did. But maybe that quality alone can be my motivation for a bit. I may be incomparable to the inspiration Dr. Sabuwala was, but this one, amazing quality of his, I can try to obtain.

Published by Paige Bright

Hi- my name is Paige Alexandria Bright. I am a rising junior at MIT interested in mathematics and philosophy. I have been writing this blog since the beginning of COVID. Lets see where this goes.

One thought on “045. Comparable

  1. oh god equality. equality is hard. i hate equality. let’s not think about how equality works

    people aren’t even a poset; there’s no good comparison function that works probably


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