Nicholas Lim, Co-founder & Lead Editor, Cogito Collective
Say you wish to walk from Singapore to Hong Kong for the perfect bowl of wonton noodles. First, you’d have to walk from Singapore to Bangkok. But to get to Bangkok, you have to first walk to Penang. To get there, again, you’d have to first walk to Kuala Lumpur. Eventually, you would realise that you would have to walk an infinite number of midway points to get from point A to point B. Since infinity is a preposterous concept, is it even possible for you to ever reach Hong Kong?
This is one of a few paradoxes proposed by Zeno of Elea (and independently by the Mohist School of Names) to support his teacher’s idea that motion is an illusion. The conclusion was based on the argument that the concept of infinity is impossible to grasp which leads to logical paradoxes. This paradox was the thorn in the sides of many philosophers and mathematicians up till the formal definition of the limit of a function by Cauchy and Weierstrass in the 19th century.
The close ties between the mathematical and philosophical concepts of infinity can be seen in this paradox and the various attempts to resolve it. In fact, the ties are even clearer in theology, where the infinite is dealt in reality. The fundamental axiom of theology is that there is a higher being that is infinite, and finite beings like us cannot comprehend the infinite. By examining infinities in both areas of knowledge, we can see the many ways in which they overlap, and similar principles in dealing with them.
Mathematical and Physical Infinity
The first proof of the mathematical infinity comes from an 1891 paper published by German mathematician Georg Cantor. One of the founders of set theory, he produced an elegant and simple proof for mathematical infinity. In fact, he even went beyond that and proved that there is an infinite number of infinities. His work was so revolutionary and profound that he was harshly and bitterly ridiculed by his contemporaries, including the illustrious Henri Poincaré and Ludwig Wittgenstein.
His elegant and celebrated proof requires no higher-level mathematics to understand at all. Firstly, we can make a list of infinitely long strings of 1s and 0s in any random order, such that all the counting numbers (0, 1, 2, 59628, etc.) can be represented in binary notation. Since there are an infinite number of counting numbers, we call this countably infinite. Now, going down the diagonal in the list, we flip the 1s to 0s and vice versa. This new number is definitely not on the list, because it differs from every single number on this list by at least one digit, and therefore we have created a number that’s outside this “infinity”. Furthermore, since there is an infinite number of permutations (it doesn’t have to be a diagonal), we have found an infinity that’s greater than the countably infinite set of numbers. Thus, we have proven that there different types of infinities that should be treated differently.
Whether you understood that proof or not, infinities are unavoidable in our theories of nature. We see them everywhere in black holes and the interactions between electrons. However, we still haven’t developed a way to deal with them cleanly (we only have workarounds). One of the reasons why we haven’t found a unified Theory of Everything is due to infinities popping up everywhere when we try to mash the two theories of general relativity and quantum mechanics together.
The usual interpretation is that infinities point to the fringes of our theories. When we hit an infinity, it signals the limits of validity for the current theory and a new one should be developed. It points out the frontiers of our knowledge, “a frontier of unknown opportunities and perils”, in JFK’s words.
Firstly, we can make a list of infinitely long strings of 1s and 0s in any random order, such that all the counting numbers (0, 1, 2, 59628, etc.) can be represented in binary notation. Since there are an infinite number of counting numbers, we call this countably infinite. Now, going down the diagonal in the list, we flip the 1s to 0s and vice versa. This new number is definitely not on the list, because it differs from every single number on this list by at least one digit, and therefore we have created a number that’s outside this “infinity”.
In most religions, infinity tends to be used as a placeholder for what cannot be fully comprehended by the human mind. They are not want for trying though, as great saints of old have struggled with the philosophical and theological implications of god’s infinity, whichever higher being they subscribe to. In fact, all of theology can be considered a study of the philosophical infinite.
One common proof for the existence of a creator of the universe also involves the concept of an infinity. Commonly known as the cosmological argument, it argues that everything in this world has a cause and effect. However, an infinite chain of causal events is impossible, as shown by the question “which came first: the chicken or the egg?”. Therefore, there has to be something outside of the system to push the first block, so that the rest of the blocks can fall. The Kalam Cosmological Argument, as this argument is known, is based on the Arabic philosopher al-Kindī’s proof that actual infinities cannot exist in the real world.
In the conception of the divine, the infinitude of god is due to human limitations in our understanding. All of god’s attributes are the ultimate extensions of humanly attributes, where the infinite manifests in the dimensions of knowledge, virtue, or might. The ineffability of god is also linked to the fact that one cannot compare finite concepts with infinite ones. One can compare the number 1 with 1,000 by finding their ratios, but no matter how large the number, its ratio to infinity makes no sense.
The similarities between the mathematical infinity and the theological infinity allow us to apply the methods used to understand each one to the other. Furthermore, with mathematics being closer to the world we live in, “it allows us to approach infinity more than any other science.” In fact, Cantor himself was convinced that his work was communicated to him through divine revelation, as a way to understand the infiniteness of the God he believed in. There is talk of math as a “universal language”, who’s to say it cannot help us unlock deeper, abstract meanings?
The existence of infinities show us the limits of our knowledge, and they undoubtedly exist in whatever models we have constructed to understand the world. The question is what we should do about them. HP Lovecraft, the founder of a genre of horror based on the unknowable, had a strange relationship with the unknown: “pleasure to me is wonder—the unexplored, the unexpected, the thing that is hidden and the changeless thing that lurks behind superficial mutability.” But despite the dangers of venturing into the unknown, human curiosity will always win out.
This example also shows the uncanny link between two seemingly different fields of knowledge. It is a case where two fields can learn from each other. From a field as rigorous as mathematics, one can examine another field as ethereal as theology. Of course, there are problems with direct application, and this essay is definitely inadequate to deal with the intricacies of the differences in definitions of infinity. The point still remains — it is increasingly important that we use mental models in one area of knowledge and apply it to another, and in the process shed light on previously untapped fields.
About the author:
Nicholas Lim is the co-founder and lead editor of Cogito. His bachelor’s thesis was on the hierarchy problem and supersymmetry, and he hopes to pursue a Ph.D in the near future.
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Featured image: Creation of Adam by Michelangelo in the Sistine Chapel
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