FOUNDATIONS OF MATHEMATICS

Mathematicians Bridge Finite-InfiniteDivideA surprising new proof is helping to connect the mathematicsof infinity to the physical world.

Wylie Beckert for Quanta Magazine

By Natalie WolchoverMay 24, 2016

W ith a surprising new proof, two young mathematicianshave 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 numberand the next, infinitely large one. Rather, it separates two kinds ofmathematical statements: “finitistic” ones, which can be provedwithout invoking the concept of infinity, and “infinitistic” ones,which rest on the assumption — not evident in nature — thatinfinite objects exist.

Mapping and understanding this division is “at the heart of

mathematical logic,” said Theodore SlamanTheodore SlamanTheodore Slaman, a professor ofmathematics at the University of California, Berkeley. Thisendeavor leads directly to questions of mathematical objectivity,the meaning of infinity and the relationship between mathematicsand physical reality.

More concretely, the new proof settles a question that has eludedtop experts for two decades: the classification of a statementknown as “Ramsey’s theorem for pairs,” or . Whereasalmost all theorems can be shown to be equivalent to one of ahandful of major systems of logic — sets of starting assumptionsthat may or may not include infinity, and which span the finite-infinite divide — falls between these lines. “This is anextremely exceptional case,” said Ulrich KohlenbachUlrich KohlenbachUlrich Kohlenbach, a professorof mathematics at the Technical University of Darmstadt inGermany. “That’s why it’s so interesting.”

In the new proofnew proofnew proof, Keita YokoyamaKeita YokoyamaKeita Yokoyama, 34, a mathematician at theJapan Advanced Institute of Science and Technology, andLudovic PateyLudovic PateyLudovic Patey, 27, a computer scientist from Paris DiderotUniversity, pin down the logical strength of — but not at alevel most people expected. The theorem is ostensibly astatement about infinite objects. And yet, Yokoyama and Pateyfound that it is “finitistically reducible”: It’s equivalent instrength to a system of logic that does not invoke infinity. Thisresult means that the infinite apparatus in can be wielded toprove new facts in finitistic mathematics, forming a surprisingbridge between the finite and the infinite. “The result of Pateyand Yokoyama is indeed a breakthrough,” said AndreasAndreasAndreasWeiermannWeiermannWeiermann of Ghent University in Belgium, whose own work on

unlocked one step of the new proof.

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Ramsey’s theorem for pairs is thought to be the most complicatedstatement involving infinity that is known to be finitisticallyreducible. It invites you to imagine having in hand an infinite setof objects, such as the set of all natural numbers. Each object inthe set is paired with all other objects. You then color each pair ofobjects either red or blue according to some rule. (The rule mightbe: For any pair of numbers A < B, color the pair blue if B < 2 ,and red otherwise.) When this is done, states that there willexist an infinite monochromatic subset: a set consisting ofinfinitely many numbers, such that all the pairs they make withall other numbers are the same color. (Yokoyama, working withSlaman, is now generalizing the proof so that it holds for anynumber of colors.)

The colorable, divisible infinite sets in are abstractions thathave no analogue in the real world. And yet, Yokoyama andPatey’s proof shows that mathematicians are free to use thisinfinite apparatus to prove statements in finitistic mathematics —including the rules of numbers and arithmetic, which arguablyunderlie all the math that is required in science — without fearthat the resulting theorems rest upon the logically shaky notion ofinfinity. That’s because all the finitistic consequences of are“true” with or without infinity; they are guaranteed to be provablein some other, purely finitistic way. ’s infinite structures“may make the proof easier to find,” explained Slaman, “but inthe end you didn’t need them. You could give a kind of nativeproof — a [finitistic] proof.”

When Yokoyama set his sights on as a postdoctoralresearcher four years ago, he expected things to turn outdifferently. “To be honest, I thought actually it’s not finitisticallyreducible,” he said.

Courtesy of Ludovic Patey and Keita Yokohama

Ludovic Patey, left, and Keita Yokoyama co-authored aproof giving the long-sought classification of Ramsey’stheorem for pairs.

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Lucy Reading-IkkandaLucy Reading-IkkandaLucy Reading-Ikkanda for Quanta Magazine

This was partly because earlier work proved that Ramsey’stheorem for triples, or , is not finitistically reducible: Whenyou color trios of objects in an infinite set either red or blue(according to some rule), the infinite, monochrome subset oftriples that says you’ll end up with is too complex aninfinity to reduce to finitistic reasoning. That is, compared to theinfinity in , the one in is, so to speak, more hopelesslyinfinite.

Even as mathematicians, logicians and philosophers continue toparse the subtle implications of Patey and Yokoyama’s result, it isa triumph for the “partial realization of Hilbert’s program,” anapproach to infinity championed by the mathematician StephenStephenStephenSimpsonSimpsonSimpson of Vanderbilt University. The program replaces an

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earlier, unachievable plan of action by the great mathematicianDavid Hilbert, who in 1921 commanded mathematicians toweave infinity completely into the fold of finitistic mathematics.Hilbert saw finitistic reducibility as the only remedy for theskepticism then surrounding the new mathematics of the infinite.As Simpson described that era, “There were questions aboutwhether mathematics was going into a twilight zone.”

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

Aristotle’s distinction suited mathematicians’ needs until the 19thcentury. Before then, “mathematics was essentiallycomputational,” said Jeremy AvigadJeremy AvigadJeremy Avigad, a philosopher andmathematician at Carnegie Mellon University. Euclid, forinstance, deduced the rules for constructing triangles andbisectors — useful for bridge building — and, much later,astronomers used the tools of “analysis” to calculate the motionsof the planets. Actual infinity — impossible to compute by itsvery nature — was of little use. But the 19th century saw a shiftaway from calculation toward conceptual understanding.Mathematicians started to invent (or discover) abstractions —above all, infinite sets, pioneered in the 1870s by the Germanmathematician Georg Cantor. “People were trying to look forways to go further,” Avigad said. Cantor’s set theory proved to bea powerful new mathematical system. But such abstract methodswere controversial. “People were saying, if you’re givingarguments that don’t tell me how to calculate, that’s not math.”

And, troublingly, the assumption that infinite sets exist led Cantordirectly to some nonintuitive discoveries. He found that infinitesets come in an infinite cascade of sizes — a tower of infinitieswith no connection to physical reality. What’s more, set theoryyielded proofs of theorems that were hard to swallow, such as the

1924 Banach-Tarski paradox, which says that if you break asphere into pieces, each composed of an infinitely densescattering of points, you can put the pieces together in a differentway to create two spheres that are the same size as the original.Hilbert and his contemporaries worried: Was infinitisticmathematics consistent? Was it true?

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

Even though they questioned the value and consistency ofinfinitistic logic, Hilbert and his contemporaries did not wish togive up such abstractions — power tools of mathematical

WikipediaWikipediaWikipedia

Amid questions over the consistency of infinitisticmathematics, the great German mathematicianDavid Hilbert called upon his colleagues to provethat it rested upon solid, finitistic logicalfoundations.

reasoning that in 1928 would enable the British philosopher andmathematician Frank Ramsey to chop up and color infinite sets atwill. “No one shall expel us from the paradise which Cantor hascreated for us,” Hilbert said in a 1925 lecture. He hoped to stay inCantor’s paradise and obtain proof that it stood on stable logicalground. Hilbert tasked mathematicians with proving that settheory and all of infinitistic mathematics is finitisticallyreducible, and therefore trustworthy. “We must know; we willknow!” he said in a 1930 address in Königsberg — words lateretched on his tomb.

However, the Austrian-American mathematician Kurt Gödelshowed in 1931 that, in fact, we won’t. In a shocking result,Gödel proved that no system of logical axioms (or startingassumptions) can ever prove its own consistency; to prove that asystem of logic is consistent, you always need another axiomoutside of the system. This means there is no ultimate set ofaxioms — no theory of everythingno theory of everythingno theory of everything — in mathematics. Whenlooking for a set of axioms that yield all true mathematicalstatements and never contradict themselves, you always needanother axiom. Gödel’s theorem meant that Hilbert’s programwas doomed: The axioms of finitistic mathematics cannot evenprove their own consistency, let alone the consistency of settheory and the mathematics of the infinite.

This might have been less worrying if the uncertaintysurrounding infinite sets could have been contained. But it soonbegan leaking into the realm of the finite. Mathematicians startedto turn up infinitistic proofs of concrete statements about naturalnumbers — theorems that could conceivably find applications inphysics or computer science. And this top-down reasoningcontinued. In 1994, Andrew Wiles used infinitistic logic to proveFermat’s Last Theorem, the great number theory problem aboutwhich Pierre de Fermat in 1637 cryptically claimed, “I havediscovered a truly marvelous proof of this, which this margin istoo narrow to contain.” Can Wiles’ 150-page, infinity-riddledproof be trusted?

With such questions in mind, logicians like Simpson havemaintained hope that Hilbert’s program can be at least partiallyrealized. Although not all of infinitistic mathematics can bereduced to finitistic reasoning, they argue that the most important

parts can be firmed up. Simpson, an adherent of Aristotle’sphilosophy who has championed this cause since the 1970s(along with Harvey FriedmanHarvey FriedmanHarvey Friedman of Ohio State University, who firstproposed it), estimates that some 85 percent of knownmathematical theorems can be reduced to finitistic systems oflogic. “The significance of it,” he said, “is that our mathematics isthereby connected, via finitistic reducibility, to the real world.”

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Almost all of the thousands of theorems studied by Simpson andhis followers over the past four decades have turned out(somewhat mysteriously) to be reducible to one of five systemsof logic spanning both sides of the finite-infinite divide. Forinstance, Ramsey’s theorem for triples (and all ordered sets withmore than three elements) was shown in 1972 to belong at thethird level up in the hierarchy, which is infinitistic. “Weunderstood the patterns very clearly,” said Henry TowsnerHenry TowsnerHenry Towsner, amathematician at the University of Pennsylvania. “But peoplelooked at Ramsey’s theorem for pairs, and it blew all that out ofthe water.”

A breakthrough came in 1995, when the British logician DavidSeetapun, working with Slaman at Berkeley, proved that islogically weaker than and thus below the third level in thehierarchy. The breaking point between and comesabout because a more complicated coloring procedure is requiredto construct infinite monochromatic sets of triples than infinitemonochromatic sets of pairs.

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“Since then, many seminal papers regarding have beenpublished,” said Weiermann — most importantly, a 2012 resultby Jiayi Liu (paired with a result by Carl JockuschCarl JockuschCarl Jockusch from the1960s) showed that cannot prove, nor be proved by, thelogical system located at the second level in the hierarchy, onerung below . The level-two system is known to befinitistically reducible to “primitive recursive arithmeticprimitive recursive arithmeticprimitive recursive arithmetic,” a set ofaxioms widely considered the strongest finitistic system of logic.The question was whether would also be reducible toprimitive recursive arithmetic, despite not belonging at thesecond level in the hierarchy, or whether it required stronger,infinitistic axioms. “A final classification of seemed out ofreach,” Weiermann said.

But then in January, Patey and Yokoyama, young guns who havebeen shaking up the field with their combined expertise incomputability theory and proof theory, respectively, announcedtheir new result at a conference in Singapore. Using a raft oftechniques, they showed that is indeed equal in logical

Lucy Reading-IkkandaLucy Reading-IkkandaLucy Reading-Ikkanda for Quanta Magazine

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strength to primitive recursive arithmetic, and thereforefinitistically reducible.

“Everybody was asking them, ‘What did you do, what did youdo?’” said Towsner, who has also worked on the classification of

but said that “like everyone else, I did not get far.”“Yokoyama is a very humble guy. He said, ‘Well, we didn’t doanything new; all we did was, we used the method of indicators,and we used this other technique,’ and he proceeded to list offessentially every technique anyone has ever developed forworking on this sort of problem.”

In one key step, the duo modeled the infinite monochromatic setof pairs in using a finite set whose elements are“nonstandard” models of the natural numbers. This enabled Pateyand Yokoyama to translate the question of the strength of into the size of the finite set in their model. “We directly calculatethe size of the finite set,” Yokoyama said, “and if it is largeenough, then we can say it’s not finitistically reducible, and if it’ssmall enough, we can say it is finitistically reducible.” It wassmall enough.

has numerous finitistic consequences, statements aboutnatural numbers that are now known to be expressible inprimitive recursive arithmetic, and which are thus certain to belogically consistent. Moreover, these statements — which canoften be cast in the form “for every number X, there existsanother number Y such that … ” — are now guaranteed to haveprimitive recursive algorithms associated with them forcomputing Y. “This is a more applied reading of the new result,”said Kohlenbach. In particular, he said, could yield newbounds on algorithms for “term rewriting,” placing an upper limiton the number of times outputs of computations can be furthersimplified.

Some mathematicians hope that other infinitistic proofs can berecast in the language and shown to be logically consistent.A far-fetched example is Wiles’ proof of Fermat’s Last Theorem,seen as a holy grail by researchers like Simpson. “If someonewere to discover a proof of Fermat’s theorem which is finitisticexcept for involving some clever applications of ,” he said,

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To Settle Infinity Dispute,a New Law of Logic

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“then the result of Patey and Yokoyama would tell us how to finda purely finitistic proof of the same theorem.”

Simpson considers the colorable, divisible infinite sets in “convenient fictions” that can reveal new truths about concretemathematics. But, one might wonder, can a fiction ever be soconvenient that it can be thought of as a fact? Does finitisticreducibility lend any “reality” to infinite objects — to actualinfinity? There is no consensus among the experts. Avigad is oftwo minds. Ultimately, he says, there is no need to decide.“There’s this ongoing tension between the idealization and theconcrete realizations, and we want both,” he said. “I’m happy totake mathematics at face value and say, look, infinite sets existinsofar as we know how to reason about them. And they play animportant role in our mathematics. But at the same time, I thinkit’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 — thelongest bridge yet between the finite and the infinite —mathematicians and philosophers are gradually moving towardanswers to these questions. But the journey has lasted thousandsof years already, and seems unlikely to end anytime soon. Ifanything, with results like , Slaman said, “the picture hasgotten quite complicated.”

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