Perhaps surprising to many, infinity comes with different sizes. While mathematicians can establish this generic result, they cannot always calculate or pinpoint different sizes of infinity. Such incapability demonstrates that mathematics is still an incomplete logical system. To me, the incompleteness of mathematics has an existential overtone echoing the finitude of human existence in contrast to people’s pursuit of eternity, meaning and purpose for their lives. That contrast, or paradox of life, reveals the pain of finite human existence, which I understood more in a creative writing workshop organised by USP. In this post, I explore how the desire to reach the infinite and the pain of finite existence inspire mathematicians and artists in their pursuits of truth and beauty. I reflect on how encountering infinity profoundly changes my perception of death, mathematics and writing.
Some infinities are bigger than other infinities. I first encountered this unintuitive conclusion when watching the teenage-romance film The Fault in Our Stars (2014). Set in contemporary America, the film is about two terminally ill young adults falling in love and cherishing their limited time together. To them, death is imminent, and oblivion is inevitable in a secular cosmos. The film ends with a poignant note. When the pair decide to rehearse their funerals for each other, the girl tearfully confesses in her eulogy that although they have had limited time, or a “very small infinity” together, she will forever be grateful for it. Even though death will soon separate them, she finds solace in their little shared time which nevertheless encompasses infinite possibilities. That is an evocative scene for me to watch because the message transcends one’s death and oblivion by alluding to the eternity of grace beyond human mortality.
Fig 1. The script of Hazel Grace, the female protagonist, in her eulogy (Note: the Mathematical reasoning is flawed)
Although I watched the film before studying mathematics at NUS, the eulogy scene and that bizaare mathematical conclusion stayed with me. I learned to formally establish the conclusion when I was a third-year student of mathematics. In my Set Theory class, I learned that there can be infinitely many sizes of infinity. The lecturer of Set Theory, Prof. Yang Yue, often cited Georg Cantor, the German mathematician who pioneers transfinite mathematics. Cantor famously claims that if mathematicians can sort infinities according to their sizes, that ordered sequence will pave their stairs to God. To Cantor, God is probably not a concrete entity or actual existence, but the feeling of being more than human and articulating the universe with the highest precision. The religious reference in Cantor’s words reveals that transfinite numbers are not merely a mathematical concern but also an existential quest expanding the limit of human knowledge and the significance of human existence.
Modern mathematicians have been working hard on calculating exact sizes of infinite sets. But instead of paving the stairs to God, mathematicians realise that Cantor’s task is beyond the mathematical system. That is, some sizes of infinity cannot be unequivocally established by standard mathematical proofs. That incapability establishes the incompleteness of mathematics, one of the most important results in the twentieth century.
While learning Set Theory, I participated in a creative writing workshop organised by USP in the same semester. The writer-in-residence was Ms. Gita Kolanad. Gita wanted us to write something without lifting our pens everyday, even if we might end up writing a personal diary. One day, I wrote about my disillusionment when learning the incompleteness of mathematics. As I wrote, the scene from The Fault in Our Stars came back to me since it was the first time I wondered about different sizes of infinity as well as the significance of my death and oblivion in a likely indifferent universe.
When I read my “diary entry” on the incompleteness of mathematics and The Fault in Our Stars, Gita asked me to remember that feeling of pain. To her, every creative writing manifests an intimate feeling of pain, and writers tell stories to express what cannot be endured or what can only be endured. A good writer always keeps her pain. That pain is not physical; it is the unease, the disillusionment and the despair of encountering life’s ultimate finiteness. With Gita’s words, I realised that The Fault in Our Stars is a story of pain, an elegy to love and death. It is not merely a story of teenage romance. It also embodies the irreconcilable conflict between human finitude and their aspiration to grasp eternal love. As Gita expressed her interests regarding transfinite mathematics, she cited a passage from the Sanskrit classic Mahabharata. “People are mortal, but they live their lives as if they are immortal”, she cited, “and that is the greatest wisdom of all time”.
Fig 3. The writing exercise where I wrote about vocabularies of pain
As I glue pieces of my experiences together, I realise that Gita’s writing workshop helped me make better sense of the film and provided an existential context for the incompleteness of mathematics. While the mathematics of infinity consists of a series of logical propositions comparing the sizes of abstract sets, the embodied phenomenology of infinity is the pain, the paradox and the struggle of living one’s mortal life as if it is immortal. Infinity is not merely a mathematical concern, it also represents an existential struggle. Hazel appeals to different sizes of infinity in her eulogy because her little shared infinity with her boyfriend gives her comfort beyond the pain of their imminent separation. I choose to study mathematics because I want to grasp something eternal in a volatile universe; the incompleteness of the subject reminds me of the pain of my finitude and the ultimate limit of my knowledge. After Gita’s creative writing workshop, I realise that human beings are storytellers; they narrate stories of love, death and hope. Most importantly, their stories and characters embody the pain of mortal existence amid the pursuit of eternal truth and beauty that defy oblivion.
Most importantly, I learn that people are all united by their desire to reach the infinite and their pain of realising its impossibility. Still, I should not just accept life’s finitude and nihilist outlook in a secular cosmos. Watching The Fault of Our Stars again while writing this reflection, I try to reconcile the irreconcilable, to search for hope beyond the finite. I realise that I, like everyone else, have been stuck in the cycle of meaning-making and being reminded of the finitude of life and knowledge. But as Hazel has taught me in her eulogy, one can still be grateful beyond death. She finds hope and resolution to the puzzle of infinity by affirming her love, wilful choices and memories. I realise that beyond my life and its ultimate oblivion, I am free to choose the person who hurts me or remembers me, the cause that I commit to, and the way I embody my pain of finite existence in my writing. In my negligible infinity shared with this world, there is still reason to embrace its infinite possibilities, to affirm the significance of my wilful choices, and to be compassionate towards the pain of other finite beings. Some infinities are larger than other infinities. To me, that mathematical conclusion now strikes as an earnest prayer.
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