This was partly because earlier work proved that Ramsey’s theorem for triples, or $latexRT_2^3$, is not finitistically reducible: When you color trios of objects in an infinite set either red or blue (according to some rule), the infinite, monochrome subset of triples that $latexRT_2^3$ says you’ll end up with is too complex an infinity to reduce to finitistic reasoning. That is, compared to the infinity in $latexRT_2^2$, the one in $latexRT_2^3$ is, so to speak, more hopelessly infinite.

Even as mathematicians, logicians and philosophers continue to parse the subtle implications of Patey and Yokoyama’s result, it is a triumph for the “partial realization of Hilbert’s program,” an approach to infinity championed by the mathematician Stephen Simpson of Vanderbilt University. The program replaces an earlier, unachievable plan of action by the great mathematician David Hilbert, who in 1921 commanded mathematicians to weave infinity completely into the fold of finitistic mathematics. Hilbert saw finitistic reducibility as the only remedy for the skepticism then surrounding the new mathematics of the infinite. As Simpson described that era, “There were questions about whether mathematics was going into a twilight zone.”

The Rise of Infinity

The philosophy of infinity that Aristotle set out in the fourth century B.C. reigned virtually unchallenged until 150 years ago. Aristotle accepted “potential infinity” — the promise of the number line (for example) to continue forever — as a perfectly reasonable concept in mathematics. But he rejected as meaningless the notion of “actual infinity,” in the sense of a complete set consisting of infinitely many elements.

Aristotle’s distinction suited mathematicians’ needs until the 19th century. Before then, “mathematics was essentially computational,” said Jeremy Avigad, a philosopher and mathematician at Carnegie Mellon University. Euclid, for instance, deduced the rules for constructing triangles and bisectors — useful for bridge building — and, much later, astronomers used the tools of “analysis” to calculate the motions of the planets. Actual infinity — impossible to compute by its very nature — was of little use. But the 19th century saw a shift away from calculation toward conceptual understanding. Mathematicians started to invent (or discover) abstractions — above all, infinite sets, pioneered in the 1870s by the German mathematician Georg Cantor. “People were trying to look for ways to go further,” Avigad said. Cantor’s set theory proved to be a powerful new mathematical system. But such abstract methods were controversial. “People were saying, if you’re giving arguments that don’t tell me how to calculate, that’s not math.”

And, troublingly, the assumption that infinite sets exist led Cantor directly to some nonintuitive discoveries. He found that infinite sets come in an infinite cascade of sizes — a tower of infinities with no connection to physical reality. What’s more, set theory yielded proofs of theorems that were hard to swallow, such as the 1924 Banach-Tarski paradox, which says that if you break a sphere into pieces, each composed of an infinitely dense scattering of points, you can put the pieces together in a different way to create two spheres that are the same size as the original. Hilbert and his contemporaries worried: Was infinitistic mathematics consistent? Was it true?

Amid fears that set theory contained an actual contradiction — a proof of 0 = 1, which would invalidate the whole construct — math faced an existential crisis. The question, as Simpson frames it, was, “To what extent is mathematics actually talking about anything real? [Is it] talking about some abstract world that’s far from the real world around us? Or does mathematics ultimately have its roots in reality?”

Even though they questioned the value and consistency of infinitistic logic, Hilbert and his contemporaries did not wish to give up such abstractions — power tools of mathematical reasoning that in 1928 would enable the British philosopher and mathematician Frank Ramsey to chop up and color infinite sets at will. “No one shall expel us from the paradise which Cantor has created for us,” Hilbert said in a 1925 lecture. He hoped to stay in Cantor’s paradise and obtain proof that it stood on stable logical ground. Hilbert tasked mathematicians with proving that set theory and all of infinitistic mathematics is finitistically reducible, and therefore trustworthy. “We must know; we will know!” he said in a 1930 address in Königsberg — words later etched on his tomb.

However, the Austrian-American mathematician Kurt Gödel showed in 1931 that, in fact, we won’t. In a shocking result, Gödel proved that no system of logical axioms (or starting assumptions) can ever prove its own consistency; to prove that a system of logic is consistent, you always need another axiom outside of the system. This means there is no ultimate set of axioms — no theory of everything — in mathematics. When looking for a set of axioms that yield all true mathematical statements and never contradict themselves, you always need another axiom. Gödel’s theorem meant that Hilbert’s program was doomed: The axioms of finitistic mathematics cannot even prove their own consistency, let alone the consistency of set theory and the mathematics of the infinite.

This might have been less worrying if the uncertainty surrounding infinite sets could have been contained. But it soon began leaking into the realm of the finite. Mathematicians started to turn up infinitistic proofs of concrete statements about natural numbers — theorems that could conceivably find applications in physics or computer science. And this top-down reasoning continued. In 1994, Andrew Wiles used infinitistic logic to prove Fermat’s Last Theorem, the great number theory problem about which Pierre de Fermat in 1637 cryptically claimed, “I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.” Can Wiles’ 150-page, infinity-riddled proof be trusted?

With such questions in mind, logicians like Simpson have maintained hope that Hilbert’s program can be at least partially realized. Although not all of infinitistic mathematics can be reduced to finitistic reasoning, they argue that the most important parts can be firmed up. Simpson, an adherent of Aristotle’s philosophy who has championed this cause since the 1970s (along with Harvey Friedman of Ohio State University, who first proposed it), estimates that some 85 percent of known mathematical theorems can be reduced to finitistic systems of logic. “The significance of it,” he said, “is that our mathematics is thereby connected, via finitistic reducibility, to the real world.”

An Exceptional Case

Almost all of the thousands of theorems studied by Simpson and his followers over the past four decades have turned out (somewhat mysteriously) to be reducible to one of five systems of logic spanning both sides of the finite-infinite divide. For instance, Ramsey’s theorem for triples (and all ordered sets with more than three elements) was shown in 1972 to belong at the third level up in the hierarchy, which is infinitistic. “We understood the patterns very clearly,” said Henry Towsner, a mathematician at the University of Pennsylvania. “But people looked at Ramsey’s theorem for pairs, and it blew all that out of the water.”

A breakthrough came in 1995, when the British logician David Seetapun, working with Slaman at Berkeley, proved that $latexRT_2^2$ is logically weaker than $latexRT_2^3$ and thus below the third level in the hierarchy. The breaking point between $latexRT_2^2$ and $latexRT_2^3$ comes about because a more complicated coloring procedure is required to construct infinite monochromatic sets of triples than infinite monochromatic sets of pairs.