The Mathematician Who Tried to Convince the Catholic Church of Two Infinities

It might have escaped lay people at the time, but for some observers the ascension of Leo XIV as head of the Catholic Church this year was a reminder that the last time a Pope Leo sat in St. Peter’s Chair in the Vatican, from 1878 to 1903, the modern view of infinity was born. Georg Cantor’s completely original “naïve” set theory caused both revolution and revolt in mathematical circles, with some embracing his ideas and others rejecting them.

Cantor was deeply disappointed with the negative reactions, of course, but never with his own ideas. Why? Because he held firm to the belief that he had a main line to the absolute—that his ideas came direct from l’intellect divino (the divine intellect). And, like the Blues Brothers Jake and Elwood, that he was on a mission from God. So when he soured on the mathematical community in 1883, he sought new audiences in Pope Leo XIII’s Catholic Church.

This was during Cantor’s later years, a time during which his mind became fouled. He developed what I call an Isaac Newton complex: a loathsome and pathological hatred for publishing that is informed by the paranoid certainty that your contemporaries are out to sabotage you. Either they are a bunch of backstabbing haters ignorant of your work, or, far worse, they are jealous of your genius and selfishly despise you because of it. (Newton himself swore off publishing for years because of criticism of his early work.)

“My own inclinations do not urge me to publish,” Cantor wrote in 1887, echoing Newton from two centuries before. “And I gladly leave this activity to others.”

For the next several years, Cantor is increasingly focused on new audiences and tries to make inroads with Catholic authorities. The 1880s are a time when the Catholic Church is becoming more interested in scientific discovery than ever before. Leo XIII, who became pope in 1878, takes a special interest in science, especially cosmology. Science is a way forward, he claims, and he maintains an astronomical observatory at the Vatican—one whose construction he personally oversees. He fills it with the best modern equipment and keeps professional astronomers on staff.

Cantor thinks the church has a lot to offer and that set theory has a lot to offer in return. He wants the Catholic church to become aware of his views because set theory is a way to understand the infinite nature of the divine—perhaps even the mind of God, reflected in math. Isn’t that worth considering?

It’s a hard sell.

Cantor shares his work with Cardinal Johannes Franzelin of the Vatican Council, one of the leading Jesuit theologians of his day. Franzelin writes Cantor a letter on Christmas Day 1885, saying he’s gratified to receive Cantor’s work. “What greatly pleases me,” he says, is that it “appears to take not a hostile, but indeed a favorable position with regard to Christianity and Catholic principles.” Having said that, Franzelin adds, Cantor’s ideas probably could not be defended and “in a certain sense, although the author does not appear to intend it, would contain the error of pantheism.”

Cantor responds in a letter just after New Year’s Day 1886, assuring Franzelin that set theory is safe for Catholicism. There are only two infinities, he says—one for God and one for humans. The Infinitum aeternum increatum sive Absolutum (eternal, uncreated, or absolute infinity) is inaccessible to humans and reserved for the divine. Then the entirely distinct Infinitum creatum sive Transfinitum (the created or transfinite infinite) is open to mere mortals.

The cardinal responds politely a few days later, ignoring the bit about the two infinities, addressing one minor point at length, thanking Cantor for his thoughtful letter, and acknowledging that, as far as he can see, “no danger for religious truths lies in your concept of the Transfinite.” Still, Franzelin adds, he’s very busy. Please don’t write to me again, he says.

Cantor makes other attempts to interest members of the Church. He writes to a Catholic priest named Ignatius Jeiler to convince him of the need for the church to consider his ideas. He contacts a Dominican priest in Rome who is carefully studying Cantor’s book Grundlagen and trying to tease out its implications. “From me, Christian philosophy will be offered for the first time the true theory of the infinite,” Cantor tells him.

His overtures to Catholic authorities would normally be strange behavior for a mathematician—especially one who is not Catholic—but they are the least weird part of his late-life weirdness. He also tries to prove that the English philosopher Francis Bacon is secretly the author of all of Shakespeare’s plays—an elitist trope that’s been around almost as long as the great bard himself. There are lots of alt-Shakespeare theories in existence—even today. The Earl of Oxford wrote his plays. Christopher Marlowe wrote them. The Earl of Derby was really the author. It’s an arrogant thesis informed by the irrational conviction that no low-born artist could ever write so well.

Even as widening cracks are appearing in Cantor’s psyche, he has his champions. German mathematician David Hilbert in particular loves Cantor’s set theory because it lends itself to something known as the pure existence proof, an invaluable tool for mathematics developed in the late 19th century that can establish the truth of a mathematical proposition without demonstrating it. Hilbert falls in love with this approach in college. He witnesses a German mathematician named Ferdinand von Lindemann winning a prestigious chair at the University of Konigsberg by using a pure existence proof to demonstrate the “transcendence” of the number pi—meaning the number is not the root of a nonzero polynomial with rational or integer coefficients. Lindemann proves that pi is a transcendental number by showing it wouldn’t make sense if it weren’t.

Hilbert takes a similar approach to proving something known as Gordon’s theorem, which for years has been elusive and seemingly unsolvable. The mathematician Paul Gordon, who develops the eponymous theorem, could prove only a special case of it. For 20 years after that, mathematicians all over Europe are unable to extend his proof beyond that simple case. Hilbert does it in a completely novel way by using a pure existence proof in 1888, which one of his students later describes as ex ungue leonem (from the lion’s claws). That cat has teeth!

Gordon initially responds by criticizing Hilbert’s revolutionary work specifically because of his pure existence proof. “This is not mathematics, but theology,” he famously says. Later, after he realizes Hilbert’s proof was correct, he concedes that point with a joke by way of apology: “I have convinced myself that theology also has its merits.”

In the 1890s, Cantor distracts himself by devoting countless hours to an organization called the Society of German Scientists and Physicians, which inspires him to create a similar organization completely devoted to math called the Deutsche Mathematiker Vereinigung (German Mathematical Society). The inaugural meeting takes place in 1891.

He sets a trap at this meeting for mathematician Leopold Kronecker by preparing his brilliantly original method of diagonalization, to prove that “large infinity,” the uncountable ℵ₁ set of real numbers, is a nondenumerable infinite set. He may have invented diagonalization solely to embarrass Kronecker. One of Cantor’s biographers implies that his ulterior motive for even founding the Mathematiker Vereinigung in the first place was to seek a stage where he could force Kronecker’s hand and goad him into open confrontation, like some 19th-century Hamlet put into a farce. Then he would proceed to embarrass him in front of everyone by countering with diagonalization and other technical arguments. Cantor’s own letters bear this out somewhat. “Many who were previously blinded would have their eyes opened for the first time,” he says, looking forward to the first meeting of the Mathematiker Vereinigung, where his trap is to be sprung—just as any good Hamlet would do. Theory or not theory!

The confrontation—if that’s indeed Cantor’s intention—never materializes, however. Kronecker’s wife is terribly injured in a mountain-climbing accident some weeks before and dies just prior to the meeting. Kronecker can’t come. He sends a short, sweet, friendly letter instead, and the assembled members of the group at the first meeting read it aloud and respond by voting him onto their board of directors. He never assumes the post, however. Kronecker himself dies a few months later.

Following Kronecker’s death, Cantor’s effort to build an international community for mathematics succeeds in a big way. The First International Congress of Mathematicians takes place in Zurich in 1897. By then, most people have come to fully appreciate the power of Cantor’s set theory, and in the opening remarks at the conference, the plenary speaker acknowledges him. Set theory is an enormous contribution, the speaker says. This inspires Cantor to jump back into mathematics with both feet in the closing years of the 19th century, and he soon publishes what his biographers call his best-known and most complete publications, two papers that lay out the subject of set theory, its principles, and its implications “in an almost perfect logical form,” the mathematician Philip Jourdain will write in 1912.

Those two papers will turn out to be his last great work. There are many more bumps in the road ahead. Major technical problems with set theory are emerging—logical problems known as paradoxes that seem to shake the integrity of set theory to its core. One is discovered in 1897 by the Italian mathematician Cesare Burali-Forti—the so-called Burali-Forti paradox, which says that if you create a set of all the possible “ordinal numbers,” which rank different sets in the bucket first, second, third, etc., then that set would contain an ordinal larger than itself, which doesn’t make sense.

Cantor himself soon finds another one, which becomes known as Cantor’s paradox. The disconcerting discovery of these paradoxes adds to his concern over his continued inability to prove the continuum hypothesis.

“My theory stands as firm as a rock,” Cantor once wrote. His mental health, not so much. Cantor would suffer grievously in the years to come, in and out of nerve clinics for the rest of his life. Yet forever convinced that God was guiding his hand, Cantor would claim at one point that he had been “logically forced” to discover set theory. “Almost against my will,” he said. He was convinced mathematicians would one day embrace set theory, and he was right. Today it’s a cornerstone of mathematical logic.