WITH A SURPRISING new proof, two young mathematicians have found a bridge across the finite-infinite divide, helping at the same time to map this strange boundary.
The boundary does not pass between some huge finite number and the next, infinitely large one. Rather, it separates two kinds of mathematical statements: “finitistic” ones, which can be proved without invoking the concept of infinity, and “infinitistic” ones, which rest on the assumption—not evident in nature—that infinite objects exist.
Mapping and understanding this division is “at the heart of mathematical logic,” said Theodore Slaman, a professor of mathematics at the University of California, Berkeley. This endeavor leads directly to questions of mathematical objectivity, the meaning of infinity and the relationship between mathematics and physical reality.
More concretely, the new proof settles a question that has eluded top experts for two decades: the classification of a statement known as “Ramsey’s theorem for pairs,” or RT22. Whereas almost all theorems can be shown to be equivalent to one of a handful of major systems of logic—sets of starting assumptions that may or may not include infinity, and which span the finite-infinite divide—RT22 falls between these lines. “This is an extremely exceptional case,” said Ulrich Kohlenbach, a professor of mathematics at the Technical University of Darmstadt in Germany. “That’s why it’s so interesting.”
In the new proof, Keita Yokoyama, 34, a mathematician at the Japan Advanced Institute of Science and Technology, and Ludovic Patey, 27, a computer scientist from Paris Diderot University, pin down the logical strength of RT22—but not at a level most people expected. The theorem is ostensibly a statement about infinite objects. And yet, Yokoyama and Patey found that it is “finitistically reducible”: It’s equivalent in strength to a system of logic that does not invoke infinity. This result means that the infinite apparatus in RT22 can be wielded to prove new facts in finitistic mathematics, forming a surprising bridge between the finite and the infinite. “The result of Patey and Yokoyama is indeed a breakthrough,” said Andreas Weiermann of Ghent University in Belgium, whose own work on RT22 unlocked one step of the new proof.
Ramsey’s theorem for pairs is thought to be the most complicated statement involving infinity that is known to be finitistically reducible. It invites you to imagine having in hand an infinite set of objects, such as the set of all natural numbers. Each object in the set is paired with all other objects. You then color each pair of objects either red or blue according to some rule. (The rule might be: For any pair of numbers A < B, color the pair blue if B < 2A, and red otherwise.) When this is done, RT22 states that there will exist an infinite monochromatic subset: a set consisting of infinitely many numbers, such that all the pairs they make with all other numbers are the same color. (Yokoyama, working with Slaman, is now generalizing the proof so that it holds for any number of colors.)
The colorable, divisible infinite sets in RT22 are abstractions that have no analogue in the real world. And yet, Yokoyama and Patey’s proof shows that mathematicians are free to use this infinite apparatus to prove statements in finitistic mathematics—including the rules of numbers and arithmetic, which arguably underlie all the math that is required in science—without fear that the resulting theorems rest upon the logically shaky notion of infinity. That’s because all the finitistic consequences of RT22 are “true” with or without infinity; they are guaranteed to be provable in some other, purely finitistic way. RT22’s infinite structures “may make the proof easier to find,” explained Slaman, “but in the end you didn’t need them. You could give a kind of native proof—a proof.”
When Yokoyama set his sights on RT22 as a postdoctoral researcher four years ago, he expected things to turn out differently. “To be honest, I thought actually it’s not finitistically reducible,” he said.
This was partly because earlier work proved that Ramsey’s theorem for triples, or RT23, 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 RT23 says you’ll end up with is too complex an infinity to reduce to finitistic reasoning. That is, compared to the infinity in RT22, the one in RT23 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? 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.