by Kate; art by Elsa Bonyhadi
I think present-day reason is the analogue to the flat earth of the medieval period. If you go too far beyond it, you’re presumed to fall off, into insanity. And people are very much afraid of that. I think this fear of insanity is comparable to the fear people once had of falling off the edge of the world. Or the fear of heresy... But the world is round, and infinitely traversable.
Insanity may be a colossal counter-evolutionary factor, a hardship that comes and drives our intelligent species to extinction before we can evolve enough to travel too far into space.
Insanity does not stand up against natural selection, so it does not come about until a species becomes intelligent enough to change the rules of natural selection, ashumans have. Animals don’t go insane in the wild, but as we’ve bred them and raised them in our cities and zoos, there have been more and more cases of animal insanity. So maybe insanity is a consequence of meddling in the world’s secrets and changing its settings. After all, brilliant people—those who meddle with the world’s secrets the most—die madly. Consider Nikola Tesla, Kurt Gödel, Ludwig Boltzmann, Alan Turing.
It’s not logic that drives geniuses up their mental walls, although that is a method that gets them there. It is abstraction that might make mathematicians go mad. The word comes from the Latin abstractus, meaning “drawn away.” Abstractions can be applied to tangible things, but they have an existence of their own, withdrawn from the particular examples. This is why numbers are so tricky. The ancient Egyptians and Babylonians both used advanced mathematical principles for construction and navigation, but, in both cases, their math was only practical. They had no concept of a number apart from the distance or objects that it represented. It was the Greeks who made math into an abstract system. Among the Greeks, it was the mathematician Pythagoras who first recognized that numbers are, in the words of mathematician Moris Kline, “ideas entertained by the mind and sharply distinguished from physical objects or pictures.”
Many of Pythagoras’ theories have stood the test of time. The Pythagorean theorem, for example, you learned in high school. But there’s an eccentric side of Pythagoras that is often overlooked. His philosophy of vegetarianism, for example, is little known. One of its tenets prohibited both touching and eating beans. And this was no idle theory: legend has it that it was the forbidden beans that led to Pythagoras’ demise. He was chased from his house to a beanfield by attackers (whether they were real and what form they took is unknown). Allegedly, the father of abstraction decided he’d rather die than enter this field and risk touching the beans.
After the death of Pythagoras, the Greeks were still lacking one matter of abstraction. They didn’t have the concept of zero, and it might have been this deficiency in their mathematical system that made the concept of infinity so impossible for them to imagine. The Hellenic term apeiron meant both infinity and “that-which-cannot-be-handled.” For example, a crumpled piece of paper was apeiron because, mathematically speaking, it was “hopelessly complex.” You can imagine how upsetting something like this would have been to a culture known for their earnest quest for knowledge. Apeiron was a pejorative term for the Greeks, but it was nevertheless firmly rooted in their mythology, as springing from creation’s original chaos.
The pre-Socratic philosopher Zeno of Elea (490 B.C.E.) is the real founder of the philosophy of the infinite. Zeno’s basic argument counters even the possibility of motion. It deals with plurality and continuity and is formally known as “the Dichotomy.” It essentially states that you have to cross an infinite number of tiny distances in order to move from one location to another, and you can’t cross an infinite number of distances in a finite amount of time. But, as you well know, you can walk from your room to the dining hall. This must mean that the Dichotomy is a paradox in theory but not in practice (which is, yes, paradoxical). Zeno’s paradoxes are important because they reveal that infinity exists in multiple ways. There is an infinity of expansion: one can count forever—and an infinity of division: one can divide forever.
A century later, Aristotle studied Zeno’s paradoxes. It’s clear that Zeno’s infinities frustrated Aristotle, who famously said, “No great mind has ever existed without a touch of madness.” Unlike Pythagoras and Plato before him, Aristotle acquiesced to the theories that seemed to point to the existence of infinity. For example, it seemed that time would go on forever and, as Zeno showed, it seemed that space could be divided into infinitely smaller sections. To avoid the threat that these prospects posed to his orderly world, Aristotle created a distinction between “actual” and “potential” infinities. There are hosts of potential infinities in the world, he admitted, but given our spatial and temporal limitations, none of these could manifest physically and completely. Therefore, no actual infinities exist.
These conceptions may have saved Aristotle from the traumatic possibility of the infinite, but they caused an epidemic of complacency in the larger society. The search for infinity effectively came to a stop for 1,700 years. Aristotle’s theories outlived him. Those ideas outlived him and were incorporated into early Church doctrine. The Christian ideology of infinity that drew on Aristotelian metaphysics became the dominant philosophical force in the West, and it claimed to have figured out infinity. This arrogance likely wouldn’t have lasted so long, had it not been swept up by the Roman Empire. Held safe by the power of the empire, in which “church” and “state” were interchangeable, the paradigm slid across the globe.
Historically, logic and faith are the two most common methods humans use to think their way around gaps in understanding. Infinity is entrenched in both of them. God is infinite in both the Old and New Testaments. Plotinus, writing in the third century C.E., had brought the belief that God is the only “actual infinity” into the world of philosophy: “[the] Absolute One has never known measure and stands outside of number, and so is under no limit in regard either to anything external or internal; for any such determinate would bring something of the dual into it.” God cannot contain paradox. This belief echoes the logic of contradiction: “this” cannot exist simultaneously with “not-this.” It was with this logic that Aristotle concluded that the paradoxical nature of infinite sets meant that infinity couldn’t be reasoned through.
Plotinus’ theory of God lasted long after the fall of Greece, arising again centuries later through the voice of Thomas Aquinas, a medieval theologian who lived during the 13th century. Aquinas agreed with Plotinus, St. Augustine and many other thinkers that God was infinite. Additionally, Aquinas thought that “although God’s power is unlimited, He still cannot make an absolutely unlimited thing, no more than He can make an unmade thing,” since this involves contradictories being true together. While the argument is well articulated, it only works on the circular premise that a “thing” is by nature finite.
In a way, the many religious thinkers on God and Infinity were the ones who immortalized Aristotle’s dual conception of infinity for so long. As no actual infinity could exist in the real world, these thinkers provided it with a place, in God, and thereby kept it from evaporating from society’s thoughts.
In Aquinas’ lifetime, the Roman Empire had dissolved and the culture was ripe for change. Even though Aquinas rejected the possibility of Actual Infinities in the created world, his attention to the question was enough to reintroduce the idea into popular thought. Aquinas provided the impetus, but it was the people who challenged the blind trust in God as the only possible “actual infinity.” For example, this paradox arose in medieval theology: How many angels can dance on the tip of a needle? The answer was contested because, given the infinite power of God, He should be able to put infinite dancing angels on the point of the needle, yet this was impossible because no actual infinity could exist in the created world.
Another medieval paradox of infinity regarded points on a circle: the circumference of a circle with a radius of one should be half as large as the circumference of a circle with a radius of two. Yet, if you draw the smaller circle circumscribed within the larger circle, every point on the circumference of the smaller circle will connect with a straight line to exactly one point on the circumference of the larger circle. The two sets of points on the circles seem to be simultaneously different, because one circle’s circumference is double the other, and equal, because both circumferences contain an infinity of points.
In the 17th century, this paradox found both name and clarity in the thinking of Galileo Galilei. Galileo’s Paradox can be mapped by showing the one-to-one correspondence between the set of natural numbers and the set of perfect squares: on one hand, a perfect square only comes along every once in a while in the set of natural numbers, so it would seem that the set of natural numbers is larger than the set of perfect squares. On the other hand, every natural number is the square root of exactly one perfect square, making it seem that the sets are equally (and infinitely) large. From this paradox, Galileo concluded, “the totality of all numbers is infinite, and the number of squares is also infinite… [therefore] the attributes ‘equal,’ ‘greater,’ or ‘less’ are not applicable to infinite but only to finite quantities.”
It is here with Galileo’s Paradox that the first signs of our modern theory of infinitesimals are visible. The theory claims that infinite sets do not behave like finite sets; they follow different rules.
Galileo, in fact, was on his way to becoming a monk before the intellectual spirit took him. He was a bold intellectual and widely ostracized for it. Although Copernicus is credited with the discovery that the Earth circles the sun, he never wrote down his theories for fear that they would jeopardize his day job at the local church. A century later, Galileo was sentenced to life imprisonment for promoting Copernicus’ theories. This sentence was then reduced, and Galileo was kept under house arrest until he died.
Galileo’s forced isolation is an outlier, but solitary house arrest (usually self-imposed) is a common plight of intellectuals. Isolation of various forms is also a common symptom of insanity. Whether Galileo was formally crazy is unknown, but his community certainly thought he was. As a man, he was obsessive and a bit odd. Among his invention sketches were plans for an automated tomato picker and a pocket comb that doubled as an eating utensil: “the hairy fork.” Once, in his solitude, Galileo mathematically determined that Dante’s Inferno measured 2000 arm-lengths.
Intellectual pursuits detract from everyday life, and as a consequence, drive intellectuals away from the empathy of others. Obsessiveness is a key factor that drives intellectuals onward, even as their distance from their community widens. Subsequently, it’s often the case that they care more for their intellectual obsessions and less for their connections with loved ones. Maybe an intellectual’s commitment to their project stems in part from a fear of it, or from a fear of not finishing it. The threat that the intellectual’s project poses is the fear of the unknown, and is thus intimately related with the fear of death.
Death is what gives life meaning, after all. It is death that drives us. “That-which-is-beyond-understanding.” The Greeks understood: apeiron, the hopelessly complex, the infinite, a pejorative word. A modern iteration, apeirophobia, captures this fear. Death is the poignant gap in our knowledge that I mentioned before. Intellectuals face apeirophobia with logic and the religious face it with faith. Gottfried Wilhelm Leibniz, a mathematician and philosopher, once said that because in his head there exists a baffling idea of infinity, he must be in this world, not alone in his head. Further, he said that God must have given him that thought as he, being a mortal and finite being, could not have conceived of such a thing alone.
Even Georg Cantor, the father of the modern theory of infinitesimals, was religious. But his religiosity was somewhat nonconventional. For Cantor, faith had called him to apeiron rather than shielding him from it. He said that ever since he was a boy, he had heard a secret voice calling him to mathematics. In his mind, this was God, and it was evidence that his discoveries were true because they came to him from what he would later call “the absolute infinity.” In fact, his last publication in 1895 began with three aphorisms, the third being: “These things which are now hidden from you will be brought into the light” (Corinthians 4:5).
Driven by this voice, he pursued a theory of the infinite. His ideas were revolutionary not because they were novel, but because of their mathematical basis. His 1872 theory of infinity is important for the same reason that Zeno’s paradox was important more than 2,000 years earlier. Cantor proved mathematically that infinity is not an all-or-nothing concept. There are, he proved, multiple sizes—which he called “powers”—of infinity. Most important is the distinction between “countable” and “uncountable” infinities. For a set to be countable, it must be possible to “order” it, to align it in a one-to-one correspondence with the natural numbers. Cantor proved that the set of rational and of algebraic numbers are both countable, but the set of real numbers is not. This means that the set of real numbers is a larger infinity than the sets of rational and algebraic numbers.
Cantor’s point of entry to the theory of infinity was through trigonometry and the natural world. He calculated the infinite unfolding of the Koch curve (shown above) and claimed that the infinite iteration of this sequence is a more accurate representation of reality (e.g. a coast line) than any finitely spikey variation of the uniform curve that calculus would purport it converges to.
It’s unclear whether Cantor’s obsession with math caused his mental problems or the other way around, but the two issues were certainly intertwined. There are a few familiar trends that preceded his first depressive episode in 1884. First, he was obsessed with his project, so much so that his honeymoon in Interlaken, Switzerland was mostly spent researching and discussing his progress on a theory. Secondly, his work was widely criticized, leading to his rejection from a job at a prestigious university. He took the criticism personally, further isolating himself by ceasing his longtime correspondence with other mathematicians and refusing to submit papers to certain journals.
Cantor never abandoned mathematics, even as his mind progressively left him. He published one particularly odd paper in 1894, in which he listed the ways that all the even numbers could be written as the sum of two primes. This had already been done by a mathematician named Christian Goldbach, which Cantor must have known. So it’s likely the paper says more about his state of mind than the Goldbach conjecture. The end of Cantor’s life was spent in a sanatorium where he is reported to have acted quite eccentrically. His daughter described him as at times “completely enraged” and then immediately “uncommunicatively silent.”
At the end of his life, Cantor was ill and malnourished. His 70th birthday party was canceled because of the war. He spent the last few years of his life continually writing his wife, asking to go home and apologizing for the love affair with infinity that caused him to leave her. Ultimately, though, we’re all having an affair with infinity in some way. It manifests when we lose faith, when we look up at the stars and feel ourselves falling into the earth, or when we attend open-casket funerals, or realize that the piece of paper will crumple differently every time, so you’ll never be able to map it. Cantor’s story, though certainly unique, was in some ways just an extreme case of a common predicament: our love affair with infinity.
Part of the Ritual Issue