Even as the finite encloses an infinite series
And in the unlimited limits appear,
So the soul of immensity dwells in minuta
And in the narrowest limits, no limits in here
What joy to discern the minute in infinity!
The vast to perceive in the small, what Divinity!
– Jakob Bernoulli
When we try to picture infinity, we usually think of something immense and unbounded, like the universe. But infinity can also be revealed in something infinitesimally small, like the numberless mathematical points on a line. Such is the case with the curious little object named in the title of Jorge Luis Borges’ short story, “The Aleph.” The story’s protagonist, also identified as Borges, describes the Aleph as “a small iridescent sphere of almost unbearable brilliance” that enables one to peer into every corner and every drawer and every page of every book in the world all at once. This window into the infinite can be glimpsed only from a certain angle while lying on one’s back in a darkened cellar and looking up at a particular step on a staircase. The house in which this tiny wonder is to be found was once occupied by Borges’ great unrequited love, Beatriz Viterbo, now long dead. Her father still lives there, along with her first cousin, Carlos Argentino Daneri, Borges’ hated literary rival. It is Daneri who first discloses the secret of the Aleph, perhaps out of spite, since it enables Borges to peer into a drawer containing compromising letters that his beloved had once written to her cousin. Inspired by what he has seen in the Aleph, Danieri reveals that he is hard at work on an epic poem describing the entire face of the planet. This ludicrous undertaking unaccountably wins a prestigious literary prize, while Borges’ most recent work goes unrecognized.
Borges wonders at the end of the story how the Aleph came by its name – whether Danieri chose it, or whether the name was somehow revealed to him in one of the numberless texts that he could absorb at a glance by looking into it. From things he has said to Borges, Danieri is aware that this first letter of the Hebrew alphabet has great symbolic significance for alchemists and Kabbalists alike. In Jewish mystical texts, the aleph represents the Ein Soph (literally “without end”), which describes the godhead before creation, without attributes yet containing all potentiality. In mathematics, the aleph is used in set theory to designate transfinite -- or actual infinite -- numbers.
The concept of infinity has long posed difficulties for both mathematicians and philosophers. It was the basis for Zeno of Elea’s famous paradoxes, by which this ancient Greek philosopher hoped to prove that nothing really moved. For example, in a foot race between Achilles and a tortoise, Achilles can never catch up to the tortoise because in the time it takes for him to close the gap with the tortoise, the animal will advance some distance farther on, and then again still farther while Achilles is closing the gap once more – and so on as the distances grow smaller and smaller and smaller into infinity. Zeno notwithstanding, the Greeks were never really comfortable with the idea of infinity, perhaps because it presented problems that could not be solved by logic or simple mathematical computation. Aristotle distinguished between potential and actual infinities, dismissing the latter as unattainable. “Since no sensible magnitude is infinite,” he said, “it is impossible to exceed every assigned magnitude; for if it were possible there would be something bigger than the heavens.”
Thereafter, owing to Aristotle’s lofty reputation, no serious thinker dared to broach the subject for many centuries, leaving the field to theologians who were presumably more conversant with what went on in the heavens. Theologians rather liked the idea of associating infinity with the divine, since it gave God an attribute that was apparently beyond mortal reach. The fact that we could even conceive of the infinite meant that God must have planted the idea in our heads; otherwise it was wholly outside our experience. At least not until mathematicians began toying with the idea once again, starting with Leibnitz and Newton, who provided a mathematical (if not philosophical) solution to Zeno’s paradoxes with their simultaneous invention of integral calculus. A still-bigger breakthrough occurred with Georg Cantor’s invention of set theory, which included a proof of the existence of actual infinite numbers as a mathematical entity. His designated mathematical symbol for sets of infinite numbers was the aleph.
A descendent of Sephardic Jews who had converted to Christianity, Cantor may or may not have been aware of the aleph’s symbolic importance in mystical Judaism. However, he certainly knew of the broader significance of the “Absolute Infinite,” which he described in overtly religious terms in correspondence with various church leaders, including Pope Leo XIII. But if ever it was his intention to bind theology and higher mathematics, his hopes were soon dashed. Christian theologians regarded Cantor’s proof as an intrusion upon God’s exclusive claim to infinitude. Meanwhile, his theory was denounced by many leading mathematicians of the day, including Henri Poincaré, who said it “should be banished from mathematics once and for all." Although now considered one of the great mathematicians of the 19th century, Cantor was regarded as a crank by many of his peers, which did nothing to improve his already fragile mental state. One could speculate that he was driven insane by delving into the mysteries of infinity. But, in fact, he suffered from what would now be diagnosed as bipolar disorder, ending his days in an asylum.
Cantor solved the problem of infinity by treating it not as many things but as one thing, as in his definition of a mathematical set as “a many which allows itself to be thought of as a one.” Zeno was getting at something similar with paradoxes intended to demonstrate that motion was a logical impossibility and therefore illusory. Zeno was part of a philosophical school based in Elea that regarded the universe as eternal and indivisible, which ruled out any appearance of change, motion or multiplicity. All is one, in other words, including the infinite and the infinitesimal. We have arrived finally at the conjunction of philosophy, mathematics and religion, to the place Cantor described as the Absolute Infinite.
But where did this idea come from? Cantor believed that it came from God and that his theory of transfinite numbers had been communicated to him directly by the Almighty – a notion that no doubt only delayed its eventual acceptance by his fellow mathematicians. Three centuries earlier, the philosopher René Descartes had also considered the origins of infinity and had arrived at the same conclusion. How otherwise could finite humans have conceived of something completely beyond them unless it had come from an infinite being? Descartes briefly considered the possibility that we are infinite beings and don’t know it, but then dismissed the idea. And yet, according to our sacred myths, we are made in God’s image, suggesting that some semblance of his infinite being may have rubbed off on us. In Borges’ short story, the Aleph can be seen only from a certain angle by one person at a time, and the world that is revealed to him is peculiarly his own. Whatever the origins of the Infinite, it may well be that we are the Aleph, the microcosm that encompasses the universe.
Zachary McCune, “United Infinity: Borges & The Aleph”
René Descartes, Discourse on Method
Peter Suber, “Infinite Reflections”