This is changing, just a bit. Since 2008, umpires have been allowed to consult video replay when they’re unsure of what actually took place on the field. This is good for getting calls right instead of wrong, but many longtime baseball fans feel it’s somehow foreign to the spirit of the sport. I’m one of them. I’ll bet John Roberts is too.
Not everybody shares Scalia’s view of the law (note that his opinion in Davis was in the minority). As we saw in Atkins v. Virginia, the words of the Constitution, like “cruel and unusual,” leave a remarkable amount of space for interpretation. If even the great Euclid left some ambiguity in his axioms, how can we expect any different from the framers? Legal realists, like judge and University of Chicago professor Richard Posner, argue that Supreme Court jurisprudence is never the exercise in formal rule following that Scalia says it is:
Most of the cases the Supreme Court agrees to decide are toss-ups, in the sense that they cannot be decided by conventional legal reasoning, with its heavy reliance on constitutional and statutory language and previous decisions. If they could be decided by those essentially semantic methods, they would be resolved uncontroversially at the level of a state supreme court or federal court of appeals and never get reviewed by the Supreme Court.
In this view, the hard questions about law, the ones that make it all the way to the Supremes, are left indeterminate by the axioms. The justices are thus in the same position Pascal was when he found he couldn’t reason his way to any conclusion about God’s existence. And yet, as Pascal wrote, we don’t have the choice not to play the game. The court must decide, whether it can do so by conventional legal reasoning or not. Sometimes it takes Pascal’s route: if reason does not determine the judgment, make the judgment that seems to have the best consequences. According to Posner, this is the path the justices finally adopted in Bush v. Gore, with Scalia on board. The decision they arrived at, Posner says, was not really supported by the Constitution or judicial precedent; it was a decision made pragmatically, in order to close off the possibility of many more months of electoral chaos.
THE SPECTER OF CONTRADICTION
Formalism has an austere elegance. It appeals to people like G. H. Hardy, Antonin Scalia, and me, who relish that feeling of a nice rigid theory shut tight against contradiction. But it’s not easy to hold to principles like this consistently, and it’s not clear it’s even wise. Even Justice Scalia has occasionally conceded that when the literal words of the law seem to require an absurd judgment, the literal words have to be set aside in favor of a reasonable guess as to what Congress must have meant. In just the same way, no scientist really wants to be bound strictly by the rules of significance, no matter what they say their principles are. When you run two experiments, one testing a clinical treatment that seems theoretically promising and the other testing whether dead salmon respond emotionally to romantic photos, and both experiments succeed with p-values of .03, you don’t really want to treat the two hypotheses the same. You want to approach absurd conclusions with an extra coat of skepticism, rules be damned.
Formalism’s greatest champion in mathematics was David Hilbert, the German mathematician whose list of twenty-three problems, delivered in Paris at the 1900 International Congress of Mathematics, set the course for much of twentieth-century math. Hilbert is so revered that any work that touches even tangentially on one of his problems takes on a little extra shine, even a hundred years later. I once met a historian of German culture in Columbus, Ohio, who told me that Hilbert’s predilection for wearing sandals with socks is the reason that fashion choice is still noticeably popular among mathematicians today. I could find no evidence this was actually true, but it suits me to believe it, and it gives a correct impression of the length of Hilbert’s shadow.
Many of Hilbert’s problems fell quickly; others, like number 18, concerning the densest possible packing of spheres, were settled only recently, as we saw in chapter 12. Some are still open, and hotly pursued. Solving number 8, the Riemann Hypothesis, will get you a million-dollar prize from the Clay Foundation. At least once, the great Hilbert got it wrong: in his tenth problem, he asked for an algorithm that would take any equation and tell you whether it had a solution in which all the variables took whole-number values. In a series of papers in the 1960s and ’70s, Martin Davis, Yuri Matijasevic, Hilary Putnam and Julia Robinson showed that no such algorithm existed. (Number theorists everywhere breathed a sigh of relief—it might have been a bit dispiriting had it transpired that a formal algorithm could autosolve the problems we’ve spent years on.)
Hilbert’s second problem was different from the others, because it was not so much a mathematical question as a question about mathematics itself. He began with a full-throated endorsement of the formalist approach to mathematics:
When we are engaged in investigating the foundations of a science, we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science. The axioms so set up are at the same time the definitions of those elementary ideas; and no statement within the realm of the science whose foundation we are testing is held to be correct unless it can be derived from those axioms by means of a finite number of logical steps.
By the time of the Paris lecture, Hilbert had already revisited Euclid’s axioms and rewritten them to remove any trace of ambiguity; at the same time, he had rigorously squeezed out any appeal to geometric intuition. His version of the axioms really does make just as much sense if “point” and “line” are replaced by “frog” and “kumquat.” Hilbert himself famously remarked, “One must be able to say at all times—instead of points, straight lines, and planes—tables, chairs, and beer mugs.” One early fan of Hilbert’s new geometry was the young Abraham Wald, who, while still a student at Vienna, had shown how some of Hilbert’s axioms could be derived from the others, and were thus expendable.*
Hilbert was not content to stop with geometry. His dream was to create a purely formal mathematics, where to say a statement was true was precisely to say it obeyed the rules laid down at the beginning of the game, no more, no less. It was a mathematics Antonin Scalia would have liked. The axioms Hilbert had in mind for arithmetic, first formulated by the Italian mathematician Guiseppe Peano, hardly seem the sort of thing about which there could be any interesting questions or controversy. They say things like “Zero is a number,” “If x equals y and y equals z, then x equals z,” and “If the number directly following x is the same as the number directly following y, then x and y are the same.” They’re the truths we hold to be self-evident.
What’s remarkable about these Peano axioms is that from these bare beginnings one can generate a great deal of mathematics. The axioms themselves seem to refer only to whole numbers, but Peano himself showed that, starting from his axioms and proceeding purely by definition and logical deduction, one could define the rational numbers and prove their basic properties.* The mathematics of the nineteenth century had been plagued by confusion and crises as widely accepted definitions in analysis and geometry were found to be logically flawed. Hilbert saw formalism as a way of starting over clean, building on a foundation so basic as to be completely incontrovertible.
But a specter was haunting Hilbert’s program—the specter of contradiction. Here’s the nightmare scenario. The community of mathematicians, working together in concert, rebuilds the entire apparatus of number theory, geometry, and calculus, starting from the bedrock axioms and laying on new theorems, brick by brick, each layer glued to the last by the rules of deduction. And then, one day, a mathematician in Amsterdam proves that a certain mathematical assertion is the case, while another mathematician in Kyoto proves that it is not.
Now what? Starting from assertions one cannot possibly doubt, one has arrived at a contradiction. Reductio ad absurdum. Do you conclude that the axioms were wrong? Or that there’s something wrong with the structure of logical deduction itself? And what do you do with t
he decades of work based on those axioms?*
Thus, the second problem among those Hilbert presented to the mathematicians gathered in Paris:
But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms: To prove that they are not contradictory, that is, that a definite number of logical steps based upon them can never lead to contradictory results.
One is tempted simply to assert that such a terrible outcome can’t happen. How could it? The axioms are obviously true. But it was no less obvious to the ancient Greeks that a geometric magnitude must be a ratio of two whole numbers; that’s how their notion of measurement worked, until the whole framework got mugged by the Pythagorean Theorem and the stubbornly irrational square root of 2. Mathematics has a nasty habit of showing that, sometimes, what’s obviously true is absolutely wrong. Consider the case of Gottlob Frege, the German logician who, like Hilbert, was laboring to shore up the logical underpinnings of mathematics. Frege’s focus was not number theory, but the theory of sets. He, too, started from a sequence of axioms, which seemed so obvious as to hardly need stating. In Frege’s set theory, a set was nothing other than a collection of objects, called elements. Often we use curly brackets {} to denote the sets whose elements are thereby enclosed; so that {1,2,pig} is the set whose elements are the number 1, the number 2, and a pig.
When some of those elements enjoy a certain property and others don’t, there’s a set that is the collection of all those elements with the specified property. To make it a little more down to earth: there is a set of pigs, and among those, the ones that are yellow form a set, the set of yellow pigs. Hard to find much to take issue with here. But these definitions are really, really general. A set can be a collection of pigs, or real numbers, or ideas, possible universes, or other sets. And it’s that last one that causes all the problems. Is there a set of all sets? Sure. A set of all infinite sets? Why not? In fact, both of these sets have a curious property: they are elements of themselves. The set of infinite sets, for example, is certainly itself an infinite set; its elements include sets like
{the integers}
{the integers, and also a pig}
{the integers, and also the Eiffel Tower}
and so on and so on. Clearly there’s no end.
We might call such a set ouroboric, after the mythical snake so hungry it chows down on its own tail and consumes itself. So the set of infinite sets is ouroboric, but {1,2,pig} is not, because none of its elements is the set {1,2,pig} itself; all its elements are either numbers or farm animals, but not sets.
Now here comes the punch line. Let NO be the set of all non-ouroboric sets. NO seems like a weird thing to think about, but if Frege’s definition allows it into the world of sets, so must we.
Is NO ouroboric or not? That is, is NO an element of NO? By definition, if NO is ouroboric, then NO cannot be in NO, which consists only of non-ouroboric sets. But to say NO is not an element of NO is precisely to say NO is non-ouroboric; it does not contain itself.
But wait a minute—if NO is non-ouroboric, then it is an element of NO, which is the set of all non-ouroboric sets. Now NO is an element of NO after all, which is to say that NO is ouroboric.
If NO is ouroboric, it isn’t, and if it isn’t, it is.
This, more or less, was the content of a letter the young Bertrand Russell wrote to Frege in June of 1902. Russell had met Peano in Paris at the International Congress—whether he attended Hilbert’s talk isn’t known, but he was certainly on board with the program of reducing all of mathematics to a pristine sequence of deductions from basic axioms.* Russell’s letter starts out like a fan letter to the older logician: “I find myself in full accord with you on all main points, especially in your rejection of any psychological element in logic and in the value you attach to a conceptual notation for the foundations of mathematics and of formal logic, which, incidentally, can hardly be distinguished.”
But then: “I have encountered a difficulty only on one point.”
And Russell explains the quandary of NO, now known as Russell’s paradox.
Russell closes the letter by expressing regret that Frege had not yet published the second volume of his Grundgesetze (“Foundations”). In fact, the book was finished and already in press when Frege received Russell’s letter. Despite the respectful tone (“I have encountered a difficulty,” not “Hi, I’ve just borked your life’s work”), Frege understood at once what Russell’s paradox meant for his version of set theory. It was too late to change the book, but he hurriedly appended a postscript recording Russell’s devastating insight. Frege’s explanation is perhaps the saddest sentence ever written in a technical work of mathematics: “Einem wissenschaftlichen Schriftsteller kann kaum etwas Unerwünschteres begegnen, als dass ihm nach Vollendung einer Arbeit eine der Grundlagen seines Baues erschüttert wird.” Or: “A scientist can hardly encounter anything more undesirable than, just as a work is completed, to have its foundation give way.”
Hilbert and the other formalists didn’t want to leave open the possibility of a contradiction embedded like a time bomb in the axioms; he wanted a mathematical framework in which consistency was guaranteed. It wasn’t that Hilbert really thought there was likely to be a contradiction hidden in arithmetic. Like most mathematicians, and even most normal people, he believed that the standard rules of arithmetic were true statements about the whole numbers, so they couldn’t really contradict one another. But this was not satisfying—it relied on the presupposition that the set of whole numbers actually existed. This was a sticking point for many. Georg Cantor, a few decades earlier, had for the first time put the notion of the infinite on some kind of firm mathematical footing. But his work had not been digested easily or accepted universally, and there was a substantial group of mathematicians who felt that any proof relying on the existence of infinite sets ought to be considered suspect. That there was such a thing as the number 7, all were willing to accept. That there was such a thing as the set of all numbers was the question at issue. Hilbert knew very well what Russell had done to Frege and was keenly aware of the dangers posed by casual reasoning about infinite sets. “A careful reader,” he wrote in 1926, “will find that the literature of mathematics is glutted with inanities and absurdities which have had their source in the infinite.” (The tone here would not be out of place in one of Antonin Scalia’s sweatier dissents.) Hilbert sought a finitary proof of consistency, one that did not make reference to any infinite sets, one that a rational mind couldn’t help but wholly believe.
But Hilbert was to be disappointed. In 1931, Kurt Gödel proved in his famous second incompleteness theorem that there could be no finitary proof of the consistency of arithmetic. He had killed Hilbert’s program with a single stroke.
So should you be worried that all of mathematics might collapse tomorrow afternoon? For what it’s worth, I’m not. I do believe in infinite sets, and I find the proofs of consistency that use infinite sets to be convincing enough to let me sleep at night.
Most mathematicians are like me, but there are some dissenters. Edward Nelson, a logician at Princeton, circulated a proof of the inconsistency of arithmetic in 2011. (Fortunately for us, Terry Tao found a mistake in Nelson’s argument within a few days.) Vladimir Voevodsky, a Fields Medalist now at the Institute for Advanced Study in Princeton, made a splash in 2010 when he said that he, too, saw no reason to feel sure that arithmetic is consistent. He and a large international group of collaborators have their own proposal for a new foundation of mathematics. Hilbert had started out with geometry, but had quickly come to see the consistency of arithmetic as the more fundamental problem. Voevodsky’s group, by contrast, argues that geometry is the fundamental thing after all—not any geometry that would have been familiar to Euclid, but a more modern kind, called homotopy theory. Will these foundations be immune to skepticism and contradiction? Ask me in twenty years. These things ta
ke time.
Hilbert’s style of mathematics survived the death of his formalist program. Even before Gödel’s work, Hilbert had made it clear he didn’t really intend for mathematics to be created in a fundamentally formalist way. That would be too difficult! Even if geometry can be recast as an exercise in manipulating meaningless strings of symbols, no human being can generate geometric ideas without drawing pictures, without imagining figures, without thinking of the objects of geometry as real things. My philosopher friends typically find this point of view, usually called Platonism, fairly disreputable; how can a fifteen-dimensional hypercube be a real thing? I can only reply that they seem as real to me as, say, mountains. After all, I can define a fifteen-dimensional hypercube. Can you do the same for the mountain?
But we are Hilbert’s children; when we have beers with the philosophers on the weekend, and the philosophers hassle us about the status of the objects we study,* we retreat into our formalist redoubt, protesting: sure, we use our geometric intuition to figure out what’s going on, but the way we finally know that what we say is true is that there’s a formal proof behind the picture. In the famous formulation of Philip Davis and Reuben Hersh, “The typical working mathematician is a Platonist on weekdays and a formalist on Sundays.”
Hilbert didn’t want to destroy Platonism; he wanted to make the world safe for Platonism, by placing subjects like geometry on a formal foundation so unshakable that we could feel as morally sturdy the whole week as we do on Sunday.
GENIUS IS A THING THAT HAPPENS
I have made much of Hilbert’s role, as is right, but there’s a risk that by paying so much attention to the names at the top of the marquee I’ll give a misimpression of mathematics as an enterprise in which a few solitary geniuses, marked at birth, blaze a path for the rest of humankind to trot along. It’s easy to tell the story that way. In some cases, like that of Srinivasa Ramanujan, it’s not so far off. Ramanujan was a prodigy from southern India who, from childhood, produced astonishingly original mathematical ideas, which he described as divine revelations from the goddess Namagiri. He worked for years completely in isolation from the main body of mathematics, with access to only a few books to acquaint him with the contemporary state of the subject. By 1913, when he finally made contact with the greater world of number theory, he had filled a series of notebooks with something like four thousand theorems, many of which are still the subject of active investigation today. (The goddess provided Ramanujan with theorem statements, but no proofs—those are for us, Ramanujan’s successors, to fill in.)
How Not to Be Wrong : The Power of Mathematical Thinking (9780698163843) Page 40
