The Paradox of Infinity

As I finished reading The Fault in Our Stars by John Green, perhaps what stood out the most was the phrase, “Some infinities are bigger than other infinities.” As Hazel Grace looked back at the time that Augustus Waters and her had left together, she had struck a reference to a seemingly enigmatic idea — that there are neither one nor two infinities, rather, there are an infinite number of infinities.

Perhaps of all the disciplines we are aquatinted with, mathematics stands to be seen as the most definitive. Governed by rules, proofs, theorems, and logic, the subject has afforded scientists a new language with which to grasp our conceptual understandings thus far. For example, the power of mathematics can be seen in research. Statistical tests to confirm hypotheses serve to sway the course of scientific relevance in a society fueled by innovation and discovery. However, the same discipline upon which discoveries in chemistry and physics are founded upon is the same discipline which has many enigmatic properties that are not explained by mathematics’ definitive nature. Today, I will be discussing one such enigma: the paradox of infinity.

An Exploration of Infinity

Infinity is the representation of the upper-limit of all real numbers, however, the differences in its behavior from real numbers’ could not be greater. One way we can see this is in the following examination: how many real numbers are between one and two? Infinite. How many real numbers are between one and three? Infinite. The interval between one and three is twice as long as that between one and two, hence, the latter infinity must be twice as large as the former, yet they are both infinity. In the former case, ∞ = ∞, which fits our logic. In the latter, however, 2∞ = ∞. In theory, infinity multiplied by any scalar should yield simply infinity itself, however, this is where we see its deviant behavior. The equation 2∞ = ∞ can be seen as simplifying to 2 = 1 (this is not exactly what happens as you cannot divide infinity by infinity, however, for the sake of Hazel Grace’s argument, let us reduce this to a comparison of the intervals’ sizes). Clearly, that is illogical and does not obey the course of mathematical logic. It is here that we see mathematics sway from its seemingly definitive nature of which we are introduced throughout our lives.

So, why does this matter? See, infinity is a concept, not a definitive mathematical rule. It is a concept assigned to explain the unfathomable bounds of real numbers. However, this concept also bounds our ability to think. Srinivasa Ramanujan (1887–1920) is regarded as one of the best mathematicians of our time. He is most notably regarded as the man who proved that the summation of all positive integers equals -1/12. Had he applied the concept of infinity to his work, that theorem would have stopped at the first step, in which the summation of all positive integers would be assigned the answer of infinity. However, he looked past this concept and was able to prove an idea that has served practical purpose in fields such as string theory.

Conclusion

The idea of infinity is abstract. It adds a layer of uncertainty to mathematics that almost makes it uncomfortable to explore. However, Ramanujan’s work begs me to wonder, is this abstraction that defies the course of real number logic simply an obstacle to understanding our world? String theory was proposed as a means to understanding our universe, however, that theory was built with the help of an idea that ignored the existence of infinity. It is almost philosophical when we ponder whether some infinities can be larger than others, yet that philosophy breaks down when we apply simple mathematics. Perhaps infinity is the bridge between the known and the unknown. Perhaps that is what Hazel Grace grappled with in her last days with Augustus Waters.