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.

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.”

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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 RT22unlocked 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 numbersA<B, color the pair blue ifB< 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 [finitistic] 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.

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 RT22 is logically weaker than RT32 and thus below the third level in the hierarchy. The breaking point between RT22 and RT32 comes about because a more complicated coloring procedure is required to construct infinite monochromatic sets of triples than infinite monochromatic sets of pairs.

Simpson considers the colorable, divisible infinite sets in RT22 “convenient fictions” that can reveal new truths about concrete mathematics. But, one might wonder, can a fiction ever be so convenient that it can be thought of as a fact? Does finitistic reducibility lend any “reality” to infinite objects — to actual infinity? There is no consensus among the experts. Avigad is of two minds. Ultimately, he says, there is no need to decide. “There’s this ongoing tension between the idealization and the concrete realizations, and we want both,” he said. “I’m happy to take mathematics at face value and say, look, infinite sets exist insofar as we know how to reason about them. And they play an important role in our mathematics. But at the same time, I think it’s useful to think about, well, how exactly do they play a role? And what is the connection?”

With discoveries like the finitistic reducibility of RT22 — the longest bridge yet between the finite and the infinite — mathematicians and philosophers are gradually moving toward answers to these questions. But the journey has lasted thousands of years already, and seems unlikely to end anytime soon. If anything, with results like RT22, Slaman said, “the picture has gotten quite complicated.”

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Mathematicians Bridge Finite-Infinite Divide | Quanta Magazine