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.
The laws of thought don't merely describe the way the human mind works; they're not just in-here rules. They're out-there rules as well. The world itself operates in accordance with them. They're the first and final arbiters of possibility—a point that can be illustrated using the fourth law of thought, the law of causality.
The year was 1996. TWA Flight 800, a passenger jet with 230 passengers and crew aboard, was flying off the coast of Long Island, New York when it vanished from air traffic control radar screens. The wreckage was soon discovered in the water. Eyewitnesses on the ground reported that the plane blew up in midair. But in the aftermath of the tragedy, there was no clear indication what had brought the plane down. Many theories were advanced. The National Transportation Safety Board (NTSB), charged with conducting the investigation, considered each theory one by one and, one by one, the NTSB ruled each one out. Conspiracy buffs thought the plane had been accidentally shot down by an American fighter jet, but none was reported in the area. Plus, the fuselage of the airliner, which was painstakingly reassembled by NTSB investigators on the ground, showed no sign of missile damage.
That fact also ruled out terrorists on the ground firing a shoulder-launched surface-to-air missile—and, in any case, the altitude of the plane put it beyond the reach of such a weapon. There was no forensic evidence of a bomb on board, so that explanation was also nixed. Every probable cause of a midair explosion was ruled out. Once the probable causes were exhausted, the NTSB turned to highly improbable, unprecedented explanations. Perhaps a sudden, catastrophic structural failure caused the plane to snap in half—except the wrecked fuselage showed no signs of metal fatigue. Or perhaps a missile had detonated near the plane, and a random fragment penetrated the fuel tank, triggering an onboard conflagration—except, again, where would the missile have come from? In the end, the NTSB concluded that the likeliest cause of the crash was an explosion in the fuel tank adjacent to the left wing. But the investigators puzzled over an ignition source; they couldn't figure out what could have set off such an explosion. They speculated that faulty wiring might have generated a spark. But there was no evidence of that. As a matter of fact, faulty-wiring-in-the-fuel-tank became the official explanation only because the NTSB investigators couldn't definitively rule it out. In the final analysis, it was the only explanation left standing.
But was that a legitimate logical move? Why couldn't the NTSB have concluded that TWA Flight 800 had exploded without a cause? What if the investigators had filed a report declaring that the aircraft was flying along as usual, with nothing wrong, and then it just blew up for no reason whatsoever? Could that have happened?
The answer, of course, is no. The idea of an effect—in this case, the sudden flaming disintegration of a passenger jet—without a cause is inconceivable in the literal sense of the word. Your mind rules it out with a degree of certainty that equals, or at minimum comes very near, the certainty with which it rules out a logical contradiction. You'll entertain the possibility that a space alien from Mars transported into the jet and struck a match in the fuel tank before you'll accept the conclusion that the plane crashed for no reason whatsoever. The Martian scenario strains credibility, but at least it's thinkable. You can imagine it. The uncaused effect scenario, on the contrary, is utterly unthinkable.
To be sure, the NTSB could've decided that there was a cause of the crash, but that it is unknown at the present time—and that it likely will remain unknown. To suppose an unknown cause is an altogether rational conclusion. It's not very satisfying, but it doesn't violate the law of causality.
The law of causality, in other words, is axiomatic. It's a sine qua non of rational thought. Along with the laws of identity, non-contradiction, and excluded middle, the law of causality forms the ground floor of knowledge, the bedrock certainty that underlies all subsequent rational certainties.
The certainty and universality of the laws of thought collide headlong with the requisites of a rational theology. For such a theology cannot skirt the problem of actual infinity: Actual infinity does not and cannot exist. Here is where Cusa's happy piety begins to turn on itself. What he stumbled upon half a millennium ago was not a metaphor for God's limitless grandeur but an intimation of God's non-existence.
The claim that actual infinity does not and cannot exist sounds strange, of course. Many things are potentially infinite. For example, a line potentially may be divided an infinite number of times, but at no point will the actual number of divisions total infinity. Likewise a line potentially may be extended an infinite distance, but at no point will its actual length stretch to infinity. To leave the world of plane geometry, future time may be thought of as another potential infinite. Revolutions of the earth around the sun, for example, can be added one by one ad infinitum, but at no point in the future, regardless how distant, will the actual number of revolutions add up to infinity. (Plus, astronomers assure us that both the earth and the sun will be lost along the way.)
The reason infinity can never be actualized is that it's a numeric plural, a hypothetical sequence of ever-increasing values. It isn't a single amount, a very large number, x, but an endless series of very large numbers: x, x+1, x+2, x+3, and so on. To say that a thing is actually infinite is thus to say that it's x and x+1 simultaneously. But x+1 is also, by definition, not-x. Saying that a thing is x and not-x simultaneously violates the law of non-contradiction. Except the law of non-contradiction cannot be violated. Remember that the universality of the laws of thought constitutes our bedrock certainty.
To illustrate this idea in a more familiar way, consider that it's impossible for a man to be two ages at the same time. Even as he's getting older, he can never be, say, 33 and 34 years old simultaneously. The instant he turns 34, he ceases to be 33. The reason is that 34 is, by definition, not-33. His age cannot be 33 and not-33 simultaneously because that would violate the law of non-contradiction. So, too, with actual infinity. Since it's a numeric plural, x and not-x, infinity cannot be actualized. An actual infinity, in other words, would be a logical contradiction, an impossibility.
Which means, first of all, that the world itself cannot have always existed. If the world had always existed, it's age at the present moment—whether measured in milliseconds or millennia—would amount to an actual infinity. As an aside, Thomas Aquinas, who perhaps saw where the logic was pointing, twisted himself into knots trying to avoid this very conclusion; he argued that since one year replaces another year, we don't need to add them up as if they existed at once. However, calculating a sum always supposes the simultaneous existence of units, whether the units are temporal (milliseconds or millennia) or physical (marbles or melons). That's the nature of arithmetic. The mental act of adding them together for the sake of measurement supposes that the units accumulate rather than replace one another. Past time, therefore, must be measured as a simultaneous existence—whether that past time stretches back to a well-known point, say, the number of years since the New York Jets won the Super Bowl, or that past time encompasses the entire duration of the world.
The law of non-contradiction tells us that the world cannot have always existed. Thus, it must have come into existence in the beginning. Furthermore, the law of causality tells us it cannot have come into existence uncaused. The existence of the world, in other words, demands a Cause.
But not so fast.
There's still that problem of actual infinity.
The beginning of the world is demonstrable since the only other possibility, a world that has always existed, isn't a possibility at all. We can rule that out because it would violate the law of non-contradiction; it would necessitate an actual infinity.
But if the world came into existence in the beginning, then the cause of its coming into existence either: 1) requires a cause of its own, or 2) has itself always existed. Which means you're still staring at the prospect of an actual infinity. Either you've got an actually infinite regress of causes (a cause of a cause of a cause—with no first cause—which, taken together, amounts to an infinite multitude of causes); or you've got a first cause of actually infinite duration (an accumulation of milliseconds or millennia—stretching back endlessly—which, taken together, amounts to an infinite multitude of temporal units).
So pick your poison: Whether you choose an infinite regress of causes, or a first cause of infinite duration, the result is an actual infinity. Except the law of non-contradiction, one of the bedrock certainties of human reason, dictates that an actual infinity cannot exist. Nevertheless, there is a choice to be made. For even though both alternatives—an infinite regress of causes, and a first cause of infinite duration—result in a violation of the law of non-contradiction, their logical consequences are strikingly different.
The first alternative, an infinite regress of causes, entails the existence of an actual infinity which includes the knowable world—since the knowable world would stand as the latest effect in the sequence. But that cannot be the case; it is not possible. To suggest the reality of that which, by definition, is not possible would result in the annihilation of rational thought. The law of non-contradiction must remain universally binding in order for rational thought to retain the distinction between assertion and denial.
Consider: If an actual infinity were possible, then a logical contradiction, any logical contradiction, would also be possible. For example, a sentient stone—a "sentient non-sentient thing"—would also be possible. But a logical contradiction is not possible. The laws of thought are the laws of possibility. If a thing can simultaneously be and not-be, then it no longer makes sense even to talk about it. Again, if I assert that Socrates is mortal, I must also deny that he's immortal—or else the initial statement had no content, no meaning. If a logical contradiction can be reconciled, then an assertion, any assertion, can be reconciled with its denial. Possibility becomes indistinguishable from impossibility.
But, of course, if you can rule out an infinite regress of causes because it would entail an actual infinity, why wouldn't the same reasoning apply always and everywhere? After ruling out an infinite regress of causes, how can you then turn around and accept a first cause with no beginning–since that would also entail an actual infinity?
That indeed is the second alternative, which is the only alternative to an infinite regress of causes: a first cause, or First Cause, that spans an actually infinite duration. Once you rule out a cause of a cause of a cause ad infinitum, you're stuck with a First Cause with no beginning. That First Cause's duration would necessarily stretch across an infinite multitude of temporal units, whether milliseconds or millennia.
There used to be a loophole for this predicament, one that hinged on an ancient philosophical distinction—found in the writings of Augustine, Boethius, and Aquinas, among others—between infinite duration and eternality. Because God exists outside of time, their argument goes, He doesn't have any duration whatsoever. He's eternal, not infinitely old. Aquinas describes eternality as "the simultaneously-whole and perfect possession of interminable life." From God's eternal perspective, the moment of creation is simultaneous with the moment of your reading these words. God doesn't look back at creation, nor forward to your finishing this essay. The two events are simultaneously present from His viewpoint, whereas, from our temporal point of view, creation far precedes the moment of your reading these words. History appears to us an ongoing movie, with events happening at different times. To God, history appears more like an all-inclusive snapshot in which past, present, and future events happen at once.
The eternality loophole remained viable until 1905 when Albert Einstein unveiled his Theory of Special Relativity. Simultaneity and duration, Einstein demonstrated, are always relative judgments. Imagine a pair of twin brothers, Jacob and Esau. Jacob cannot bear their constant squabbling, so he takes off in a rocket that travels at almost the speed of light; Esau stays behind on earth. Twenty years pass on earth—which means 20 years pass from Esau's perspective. Then Jacob's rocket returns to earth. But from Jacob's perspective, only 10 years have passed. Esau is now 10 years older than Jacob. They're still twins; they were still born the same day. Neither of them is wrong about his age. It's just that the duration of Esau's life is now 10 years longer than the duration of Jacob's life. The notion that time slows down at high speeds is counterintuitive, but the phenomenon has been verified many times by experiment. Atomic clocks mounted inside high-speed jets actually lose miniscule fractions of a second by the time the jets land. Duration is indeed relative.
How does that bear on our present discussion? Consider: Even if, from God's eternal frame of reference, the notion of duration is meaningless, from our perspective, from the perspective of the temporal world, duration is real. God might be eternal to Himself; to us, His eternality amounts to infinite age. Einstein taught us that measurement of duration depends on the frame of reference of the measurer.
Einstein, to be sure, was talking about objects located in space and time. So it might be argued that the concepts of space and time cannot be applied to God's eternal existence. That, remember, is how Aquinas tried to dodge the problem of an actually infinite duration—if eternal existence doesn't admit change, then it doesn't admit time. But what the theory of special relativity does is noose up that loophole. Whether or not God experiences time is irrelevant. What's relevant is that we do. Time might not move from past to present to future from God's eternal perspective, but it does from ours. From our perspective, therefore, God does have duration. The theory of special relativity, moreover, dictates that our temporal perspective is every bit as substantial, every bit as real, as His. So if God has no beginning, which is the only way to avoid an actually infinite regress of causes, His duration must be actually infinite. (Again, from our frame of reference.) Right now, at this very moment, measured in milliseconds or millennia, God's duration must be actually infinite.
Except an actual infinity cannot be: It violates the law of non-contradiction. By definition, it is not possible.
We know that an actual infinity violates the law of non-contradiction. For that reason, we can rule out an actually infinite regress of causes—in other words, a cause of a cause of a cause ad infinitum. But that means we're stuck with a First Cause of actually infinite duration.
Like an infinite regress of causes, an infinite multitude of temporal units entails a logical impossibility. So at first glance the second alternative would seem no more satisfactory than the first if the actual infinity of temporal units were located within the realm of possibility.
Therein lies the wiggle room.
You can have an actual infinity outside the realm of possibility. You can locate a First Cause of infinite duration there. But "locate" must not be taken in the literal sense of the word. That First Cause, which, for the sake of convenience I'll call God, cannot be located in space or time or, as it turns out, possibility. Even if God cannot be spatially or temporally located, however, He can be conceptually located. He can be conceptually located outside the realm of possibility. Now the "realm of possibility" sounds rather like a Disney Kingdom. But in truth it's the most comprehensive logical category. The realm of possibility is broader than the physical world—that is, the totality of stuff—for it encompasses not only whatever is but also whatever can be. Indeed, the realm of possibility extends as far as the verb to be, as far as predicates attach to their subjects, as far as what's possible can be distinguished what's not possible.
The point is that an infinite accumulation of temporal units, an infinite accumulation of any sort, constitutes an actual infinity—which is not possible. It therefore cannot be since it would introduce an impossibility into the realm of possibility. However, there's no need to locate (again, so to speak) this singular case of an actual infinity within the realm of possibility. If we locate the Cause of the world outside the realm of possibility, that Cause becomes the Impossibility that accounts for possibility, the Nothing by which and out of which all things emerged.
Nothing, as in no thing.
However, to recognize (another inconvenient word) the Cause of the world as a logical impossibility, or rather to absent Him from the realm of possibility, is also to detach Him from rational language. We cannot speak of an actual infinity as if it were bound by the laws of thought. Actual infinity by definition defies the laws of thought. It is impossible. If the realm of possibility includes whatever is or can be, impossibility includes only whatever is not and cannot be. The latter qualification, that which "cannot be," is crucial. Impossibility consists entirely of non-being, of the impossible reconciliation of contradictories. Impossibility, in short, consists of nothing.
Here, then, is what atheism got right: God is nothing.
That is, God is no thing in the narrow sense that "thing-ness," the starting point of every rationally meaningful statement, and the baseline quality of existence, cannot be stipulated of Him. So He must be conceived as inconceivable. Posited as impossible. God, in short, does not exist. That which is no thing is not and cannot be. The realm of possibility excludes nothing—literally. Nothing is beyond the laws of thought, which circumscribe possibility, and which are invoked in every assertion or denial. Thus: God.
But without the law of non-contradiction, you cannot make true (or untrue) statements about God; they have no logical traction. You can neither assert nor deny. This was the insight that Cusa intuited but did not, and perhaps could not, confront. If I assert that God is just, I must also deny that God is unjust—or else the initial assertion was empty. But once a subject has been absented from the realm of possibility, any assertion can be simultaneously denied; whatever God is, He is not. Just and unjust. Loving and unloving. Eternal and not eternal, since God's eternality is reconcilable with its logical contradiction. God, in essence, can "be" both eternal and not eternal at once—which effectively voids the initial assertion that He is eternal. Whatever is said is simultaneously unsaid. That "is" (and even the word is here must be qualified) the nature of the non-being that precedes being, the Nothing or Nihil that caused the world.
Here, then, is what atheism got wrong: "In the beginning, God created the heavens and the earth."
That is, the first line of Genesis can be deduced from the laws of thought. The world cannot be infinitely old; therefore, it must have come into existence at a definite point in the past. The world cannot have come into existence without a cause; therefore, it requires an infinite Cause to account for its existence. But an actual infinity violates the laws of thought and cannot exist; therefore, even though God does not exist, God created the world. Wrestle, for a moment, with that paradox: God does not exist . . . and God created the world. The two propositions seem irreconcilable; indeed, they seem mutually exclusive. But each one is dead certain, as dead certain as the laws of thought themselves.
It is a paradox that tells us as much about the limits of human reason and rational language as it does about God. Just as a sentient stone is not a thing because its definition asserts what it denies, so too God, absented from the realm of possibility, is not a thing. God's nothingness would permit the simultaneous assertion and denial of any predicate—for example, His "omniscient ignorance" or "omnipotent powerlessness" or "omnipresent absence."
That doesn't mean that thinking and talking about God, as human beings have been doing for millennia, is pointless. It's just not logically binding. Reason must continue to regard God as a thing—though, again, thing-ness supposes possibility. As Aquinas himself notes, in the relationship between being and non-being, human reason "apprehends non-being as an extreme." That's another way of saying that the mind attributes being to non-being (or, if you prefer, thing-ness to nothingness) in the process of thinking and talking about it; we superimpose possibility on impossibility. Logically, this is illegitimate. Practically, it's unavoidable. The instant we begin to speak about God, the instant the word "is" enters the discussion, we engage absurdities. Assertions blur into their own denials. Whatever God is, He isn't. The verb "to be" has been stretched too far.
That isn't where theology ends, of course.
But it is where rational theology ends.
Mark Goldblatt teaches religious history at Fashion Institute of Technology of the State University of New York. His latest novel, Sloth, was published by Greenpoint Press last year.
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