Nicholas of Cusa (1401-1464) isn’t a household name, even among households steeped in the Judeo-Christian theological tradition. He tends to get lost in the shuffle—dwarfed by the late medieval giants who preceded him such as Maimonides, Aquinas, and Ockham, and obscured by the Reformation apologists who followed such as Luther, Calvin, and Hooker. Yet Cusa’s mystical intuitions about the nature of God are among the most eerie and profound ever put to paper. Inadvertently, though, he may have sounded the death knell of rational theology.
To make sense of God, Cusa turned not only to holy scripture but to plane geometry. Here is where his main contribution lies. The axioms of geometry forced him to wrestle with the mind-boggling difficulties of infinity. Cusa began by supposing that God must be infinite—in his words, the “Absolute Maximum.” It was a traditional notion in Cusa’s time. Saint Anselm, four centuries earlier, had described God as “that than which none greater can be thought.” But Cusa pushed the idea of God’s infinite nature farther, zeroing in on the logical paradoxes that resulted from actual infinity.
Think of a circle, he said, and then think of a straight line. By definition, the circle is not a line, and the line is not a circle. Now suppose that you’re sitting on a sand beach, with a wood stick in your hand, about to draw the circumference of a circle that is exactly one foot in diameter. So you start at the bottom, and you curl upwards after only a moment—after all, the circle is only one foot in diameter. If you don’t start curling upwards soon enough, the circle will wind up too large. On the other hand, if the diameter of the circle you’re about to draw is 10 feet, your upward curl will be more gradual. It’s going to take longer. You’re going to have to stand up and walk the stick around the circumference of the circle. The larger the diameter of the circle you are about to draw, the slower your upward curl is going to be. To draw a circle with a 100 foot diameter, you’re going to have to drag the stick through the sand with an upward curl so gradual it will seem at first almost indiscernible.
Now think of an infinite circle, a circle whose diameter equals infinity. If the diameter of the circle is infinite, think what that would do to the circumference of the circle. If you try to draw an infinite circle in the sand, starting at the bottom, you’ll never even begin to curve upwards. For the moment you begin to curve upwards, you limit the diameter of the circle—you render it finite.
Except that if you never begin to curve upwards, but just go on and on, you’re drawing a straight line, not a circle. You will go on and on towards infinity in a straight line without ever curving upwards. That’s how Cusa came to the conclusion that an infinite circle is an infinite line. By definition, of course, a circle is not a line. But at infinity, a circle is what it is not. Only at the point of infinity, Cusa argues, are contradictories reconciled.
It was at the point of infinity that Cusa found God. God is the Coincidentia Oppositorum—the Coincidence of Opposites. He is where things become what they are not. “God is the absolute maximumness and absolute unity,” Cusa writes, “preceding and uniting things that are absolutely different and distant, for example, contradictories, between which there is no mean.” In other words, God is the point at which contradictions merge into identities, at which is and is not become one.
The infinite nature of God cannot be grasped by the logic of the finite human mind, according to Cusa, but it can be glimpsed by way of another geometric metaphor. Imagine a circle, not an infinite circle this time but a run-of-the-mill circle. Now think of a square inside the circle. The square and the circle are, obviously, different. The square has four equal sides, and the circle has no sides. That’s the reason you can never square a circle; you’d have to draw the circle with sides. But a circle with sides is a contradiction in terms; it’s nonsensical. If it has sides, it’s not a circle.
Now, again, think of that square inside that circle. Except now, in your mind, add a side to the square. Make it a pentagon, with five sides, instead of a square with four. So now you are imagining a pentagon inside a circle. Notice that the pentagon looks more like the circle than the square did.
Now imagine an octagon, which has eight sides. Imagine it inside the circle. It would look even more like the circle than the pentagon did; a dodecagon, which is a 12-sided plane figure, would look even more like the circle than the octagon did. Notice that the more sides you add to the plane figure inside the circle, the more closely it comes to resemble the circle.
But a circle has no sides.
The greater the number of sides, the nearer you come to zero sides. If you imagine a plane figure with a million sides, it becomes almost indistinguishable from a circle with zero sides. Yet it’s still not a circle. It never quite becomes a circle until the number of sides becomes infinite. But the number of sides can never become actually infinite. Infinity, by definition, cannot be reached. It cannot be actualized. You can get closer and closer, but you can never quite finish it off.
Which is Cusa’s point. The act of adding sides to a plane figure brings you closer to both an infinite number of sides and to zero sides. Infinity, therefore, is the unachievable, inconceivable moment at which contradictory extremes are unified. The moment at which the greatest number and the least number become one and the same.
That’s also, according to Cusa, how the finite human mind glimpses, but does not grasp, the infinite nature of God. Rational thought and language fail at infinity because contradictories become identities. Human reason cannot grasp God, Cusa believes, because it is crippled by the law of non-contradiction. “Since reason cannot leap over contradictories,” he writes, “there is, in accord with reason’s movement, no name to which another is not opposed.” The fact that the rational mind thinks in terms of finite oppositions—Socrates either is or is not mortal—rather than in terms of infinite unities keeps us from full comprehension of the ultimate Truth.
Cusa, who oozed piety in every sentence, regarded his geometric analogies as signs of God’s infinite glory. Despite his intentions, however, he was laying the groundwork for an airtight proof of God’s nonexistence.
That proof continues with a brief tour of the basic laws of thought. There are four: the law of identity (a thing is whatever it is); the law of non-contradiction (a thing cannot simultaneously be and not be); the law of excluded middle (a thing must either be or not be), and the law of causality (for every condition or event, there must be a cause). Despite the occasional yelps from multiculturalists and postmodernists, the laws of thought are universal. No human being has ever lived, no human society has ever existed, that did not accept and rely upon the validity of the laws of thought; they are the foundation of reasoning and knowing. Descartes, for example, held that epistemology began with the proposition “I think, therefore I am.” Yet even that rests upon three prior suppositions. Descartes took for granted that he couldn’t exist and not exist simultaneously. That means he presupposed the law of non-contradiction. He also took for granted that he either had to exist or not exist, one or the other. That means he presupposed the law of excluded middle. So, too, he took for granted that he was who and what he was—that the I of “I think” was the same as the I of “I am.” That means he presupposed the law of identity. Even a proposition as basic as “I think, therefore I am” invokes three of the first four laws of thought.