As a child I had a recurring dream. I was chasing someone important — can’t remember who — but my life depended on reaching them. I kept getting closer, but not close enough.
If they were 100m ahead of me. I would make up 50m, then 25m, then 12.5m, then 6.25m.
You get the idea. I was dreaming an infinite sequence. I would always get closer, but my goal was forever out of reach.
Luckily for me I grew out of this tortuous experience, but I can’t help but wonder if my recurring dream was a premonition of my obsession with the infinite. I can’t shake it. Everywhere I look, infinity is screaming out like a misunderstood child. I take solace that I am not the first to be haunted by that beast.
Deep in a depressing, communist era high-rise development in Halle, Germany you will find the most unusual of structures — a metal cuboid, mostly ignored nowadays. Few passers-by know that this structure is a monument to the man who made the greatest progress in our understanding of the infinite, and a man who was my prime motivation to pursue the study of mathematics.
Georg Cantor arguably has left the most profound legacy in the history of Mathematics in giving us the logical tools to comprehend infinite quantity, and to differentiate between different forms of infinity. He sacrificed his reputation and his mental health in the process, but he believed it was worth it. So do I.
Infinity touches everything we do, and the more we can understand it and comprehend it, the more we can advance our progress in pretty much every discipline known to man.
In Mathematics, a greater understanding of the infinite led to incredible breakthoughs in logic, allowing us to advance the study of set theory and plug logical gaps needed to prove critical theorems through the proposition of the Axiom of Choice. Many will argue, not unreasonably, that the entire field of logic theory exists today because of our early studies of infinite sets.
In the Physical Sciences, infinity pops up as theoretical phenomena which are proposed to exist but which we have yet to explore. Discontinuities like black holes are founded on the principles of infinite asymptotes, and the debate about the boundaries of our universe liberally uses the word ‘infinite’ in a way that confuses us and tempts us at the same time.
In Philosophy, infinity and its study gives rise to existential questions about the role of mankind in the universal order. It also opens up a clear lens on the relationship between the proof of a proposition and the agreed basic rules or axioms of logic. In fact, debates in the Mathematics community about the introduction of the Axiom of Choice were more philosophical than mathematical, with some camps opening that old chestnut: when does a truth needed to be stated explicitly or when can it just be assumed as self-evident?
In Theology, an understanding of the infinite is touted as the ultimate ascension to a God-like understanding and control of the universe. In some religions, the infinite is associated uniquely with God, and mortals are discouraged from attempting to engage with such a divine concept. In fact, many of Georg Cantor’s troubles in life stemmed from his clashes with the German church, who discouraged his attempts to comprehend a concept that they viewed as the undisputed territory of the all-seeing, all-knowing creator.
Even in art, there is a beauty to the mathematics of infinity. Cantor’s proof that the set of all fractions could be enumerated (ie written in a ordered way), and therefore was no bigger than the set of whole numbers, is a thing of beauty — a simple diagram which showed how an infinite set of infinite lines could be reconfigured into a single infinite line. It’s still the simplest, most beautiful proof I have ever seen to this day. A close runner up is his proof that the continuum of all numbers between 0 and 1 cannot be enumerated (see here).
In the end, Cantor’s work brought us into a new plane of understanding of the infinite, but was not sufficient to bring it completely into our grasp. One of the consequences of his visionary mathematics was his discovery that there are infinitely many sizes of infinity — that given any infinite set, you can define a larger set that is ‘more infinite’. Perhaps this discovery means that infinity will always be that one step ahead of us, just like in my old dreams. But I never stopped the chase, and I don’t intend to now.