You might think it’s obvious that figuring out what’s true is always our proper business. But that’s not always the case in criminal law, where the difference presents itself quite starkly in the form of defendants who committed the crime but who cannot be convicted (say, because evidence was obtained improperly) or who are innocent of the crime but are convicted anyway. What’s justice here—to punish the guilty and free the innocent, or to follow criminal procedure wherever it leads us? In experimental science, we’ve already seen the dispute with R. A. Fisher on one side and Jerzy Neyman and Egon Pearson on the other. Are we, as Fisher thought, trying to figure out which hypotheses we should actually believe are true? Or are we to follow the Neyman-Pearson philosophy, under which we resist thinking about the truth of hypotheses at all and merely ask: Which hypotheses are we to certify as correct, whether they’re really true or not, according to our chosen rules of inference?
Even in math, supposedly the land of certainty, we face these problems. And not in some arcane precinct of contemporary research, but in plain old classical geometry. The subject is founded on Euclid’s axioms, which we wrote down in the last chapter. That fifth one—
If P is a point and L is a line not passing through P, there is exactly one line through P parallel to L.
—is a bit funny, isn’t it? It’s somehow a bit more complicated, a bit less obvious, than the rest. That’s how geometers saw it for centuries, at any rate.* Euclid himself is thought to have disliked it, proving the first twenty-eight propositions in the Elements using only the first four axioms.
An inelegant axiom is like a stain in the corner of the floor; it doesn’t get in your way, per se, but it’s maddening, and one spends an inordinate amount of time scrubbing and scouring and trying to make the surface nice and clean. In the mathematical context, this amounted to trying to show that the fifth axiom, the so-called parallel postulate, followed from all the others. If that were so, the axiom could be removed from Euclid’s list, leaving it shiny and pristine.
After two thousand years of scrubbing, the stain was still there.
In 1820, the Hungarian noble Farkas Bolyai, who had given years of his life to the problem without success, warned his son János against following the same path:
You must not attempt this approach to parallels. I know this way to the very end. I have traversed this bottomless night, which extinguished all light and joy in my life. I entreat you, leave the science of parallels alone. . . . I was ready to become a martyr who would remove the flaw from geometry and return it purified to mankind. I accomplished monstrous, enormous labors; my creations are far better than those of others and yet I have not achieved complete satisfaction. . . . I turned back when I saw that no man can reach the bottom of this night. I turned back unconsoled, pitying myself and all mankind. Learn from my example. . . .
Sons don’t always take advice from their fathers, and mathematicians don’t always quit easily. The younger Bolyai kept working on the parallels, and by 1823 he had formed the outline of a solution to the ancient problem. He wrote back to his father, saying:
I have discovered such wonderful things that I was amazed, and it would be an everlasting piece of bad fortune if they were lost. When you, my dear Father, see them, you will understand; at present I can say nothing except this: that out of nothing I have created a strange new universe.
János Bolyai’s insight was to come at the problem from behind. Rather than try to prove the parallel postulate from the other axioms, he allowed his mind the freedom to wonder: What if the parallel axiom were false? Does a contradiction follow? And he found that the answer was no—that there was another geometry, not Euclid’s but something else, in which the first four axioms were correct but the parallel postulate was not. Thus, there can be no proof of the parallel postulate from the other axioms; such a proof would rule out the possibility of Bolyai’s geometry. But there it was.
Sometimes, a mathematical development is “in the air”—for reasons only poorly understood, the community is ready for a certain advance to come, and it comes from several sources at once. Just as Bolyai was constructing his non-Euclidean geometry in Austria-Hungary, Nikolai Lobachevskii* was doing the same in Russia. And the great Carl Friedrich Gauss, an old friend of the senior Bolyai, had formulated many of the same ideas in work that had not yet seen print. (When informed of Bolyai’s paper, Gauss responded, somewhat ungraciously, “To praise it would amount to praising myself.”)
To describe the so-called hyperbolic geometry of Bolyai, Lobachevskii, and Gauss would take a little more space than we have here. But as Bernhard Riemann observed a few decades later, there is a simpler non-Euclidean geometry, one that’s not a crazy new universe at all: it is the geometry of the sphere.
Let’s revisit the first four axioms:
There is a Line joining any two Points.
Any Line segment can be extended to a Line segment of any desired length.
For every Line segment L, there is a Circle which has L as a radius.
All Right Angles are congruent to each other.
You might notice I’ve made a small typographical change, capitalizing the geometric terms point, line, circle, and right angle. I’ve done this, not to give the axioms an old-timey look on the page, but to emphasize that, from a strictly logical point of view, it doesn’t matter what “points” and “lines” are called; they could be called “frogs” and “kumquats” and the structure of logical deduction from the axioms should be just the same. It’s just like Gino Fano’s seven-point plane, where the “lines” don’t look like the lines we learned in school, but it doesn’t matter—the point is that they act like lines so far as the rules of geometry are concerned. It would be better, in a way, to call points frogs and lines kumquats, because the point is to free ourselves from preconceptions about what the words Point and Line really mean.
Here is what they mean in Riemann’s spherical geometry. A Point is a pair of points on the sphere which are antipodal, or diametrically opposite each other. A Line is a “great circle”—that is, a circle on the sphere’s surface, and a Line segment is a segment of such a circle. A Circle is a circle, now allowed to be of any size.
With these definitions, Euclid’s first four axioms are true! Given any two Points—that is, any two pairs of antipodal points on the sphere—there is a Line—that is, a great circle—that joins them.* What’s more (though this is not one of the axioms) any two Lines intersect in a single Point.
You might complain about the second axiom; how can we say that a Line segment can be extended to any length, when it can never be longer than the length of a Line itself, which is the circumference of the sphere? This is a reasonable objection, but comes down to a question of interpretation. Riemann interpreted the axiom to mean that lines were boundless, not that they were of infinite extent. Those two notions are subtly different. Riemann’s Lines, which are circles, have finite length but are boundless; one can travel along them forever without having to stop.
But the fifth axiom is a different story. Suppose we have a Point P, and a Line L not containing P. Is there exactly one Line through P that is parallel to L? No, for a very simple reason: in spherical geometry, there are no such things as parallel lines! Any two great circles on a sphere must intersect.
ONE-PARAGRAPH PROOF: Any great circle C cuts the sphere’s surface into two equal parts, each one of which has the same area; call this area A. Now suppose another great circle, C’, is parallel to C. Since it doesn’t intersect with C, it must be entirely enclosed on one side or the other of C, on one of those two area-A half-spheres. But this means that the area enclosed by C’ is less than A, impossible, since every great circle encloses area exactly A.
So the parallel postulate fails, in spectacular fashion. (In Bolyai’s geometry, the situation is just the opposite: there are too many parallel lines, not two few, and in fact there are infinitely many lines through P parallel to
L. As you can imagine, this geometry is a bit harder to visualize.)
If that strange condition, where no two lines are ever parallel, sounds familiar, it’s because we’ve been here before. It’s just the same phenomenon we saw in the projective plane, which Brunelleschi and his fellow painters used to develop the theory of perspective.* There, too, every pair of lines met. And this is no coincidence—one can prove that Riemann’s geometry of Points and Lines on a sphere is the same as that of the projective plane.
When interpreted as statements about Points and Lines on a sphere, the first four axioms are true but the fifth is not. If the fifth axiom were a logical consequence of the first four axioms, then the existence of the sphere would present a contradiction; the fifth axiom would be both true (by virtue of the truth of the first four axioms) and not (by virtue of what we know about spheres). By the good old reductio ad absurdum, this means that spheres do not exist. But spheres do exist. So the fifth axiom cannot be proved from the first four, QED.
This might seem like a lot of work to get a stain off the floor. But the motivation for proving statements of this kind is not just an obsessive attention to aesthetics (though I can’t deny those feelings play a role). Here’s the thing; once you understand that the first four axioms apply to many different geometries, then any theorem Euclid proves from only those axioms must be true, not only in Euclid’s geometry, but in all the geometries where those axioms hold. It’s a kind of mathematical force multiplier; from one proof, you get many theorems.
And these theorems are not just about abstract geometries made up to prove a point. Post-Einstein, we understand that non-Euclidean geometry is not just a game; like it or not, it’s the way space-time actually looks.
This is a story told in mathematics again and again: we develop a method that works for one problem, and if it is a good method, one that really contains a new idea, we typically find that the same proof works in many different contexts, which may be as different from the original as a sphere is from a plane, or more so. At the moment, the young Italian mathematician Olivia Caramello is making a splash with claims that theories governing many different fields of mathematics are closely related beneath the skin—if you like technical terms, they are “classified by the same Grothendieck topos”—and, that, as a result, theorems proved in one field of mathematics can be carried over for free into theorems in another area, which on the surface appear totally different. It’s too early to say whether Caramello has truly “created a strange new universe,” as Bolyai did—but her work is very much in keeping with the long tradition in mathematics of which Bolyai was a part.
The tradition is called “formalism.” It’s what G. H. Hardy was talking about when he remarked, admiringly, that mathematicians of the nineteenth century finally began to ask what things like
1 − 1 + 1 − 1 + . . .
should be defined to be, rather than what they were. In this way they avoided the “unnecessary perplexities” that had dogged the mathematicians of earlier times. In the purest version of this view, mathematics becomes a kind of game played with symbols and words. A statement is a theorem precisely if it follows by logical steps from the axioms. But what the axioms and theorems refer to, what they mean, is up for grabs. What is a Point, or a Line, or a frog, or a kumquat? It can be anything that behaves the way the axioms demand, and the meaning we should choose is whichever one suits our present needs. A purely formal geometry is a geometry you can in principle do without ever having seen or imagined a point or a line; it is a geometry in which it’s irrelevant what points and lines, understood in the usual way, are actually like.
Hardy would certainly have recognized Condorcet’s anguish as perplexity of the most unnecessary kind. He would have advised Condorcet not to ask who the best candidate really was, or even who the public really intended to install in office, but rather which candidate we should define to be the public choice. And this formalist take on democracy is more or less general in the free world today. In the contested 2000 presidential election in Florida, thousands of Palm Beach County voters who believed they were voting for Al Gore in fact recorded votes for the paleoconservative Reform Party candidate Pat Buchanan instead, thanks to the confusingly designed “butterfly ballot.” Had Gore received those votes instead, he would have won the state, and the presidency.
But Gore doesn’t get those votes; he never even seriously argued for them. Our electoral system is formalist: what counts is the mark made on the ballot, not whatever feature of the voter’s mind we may take it to indicate. Condorcet would have cared about the voter’s intent; we, at least officially, do not. Condorcet would have cared, too, about the Floridians who voted for Ralph Nader. Presuming, as seems safe, that most of those people preferred Gore to Bush, we see that Gore is the candidate who Condorcet’s axiom declares the victor: a majority preferred him to Bush, and an even greater majority preferred him to Nader. But those preferences aren’t relevant to the system we have. We define the public will to be that mark that appears most frequently on the pieces of paper collected at the voting booth.
Even that number, of course, is open to argument: How do we count a partially punched ballot, the so-called hanging chad? What to do with votes mailed from overseas military bases, some of which couldn’t be certified as having been cast on or before Election Day? And to what extent were Florida counties to recount the ballots in an attempt to get as precise a reckoning as possible of the actual votes?
It was this last question that made its way to the Supreme Court, where the matter was finally decided. Gore’s team had asked for a recount in selected counties, and the Florida Supreme Court had agreed, but the U.S. Supreme Court said no, fixing the total in place with Bush holding a 537-vote lead, and granting him the election. More counting would presumably have resulted in a more accurate enumeration of the votes; but this, the court said, is not the overriding goal of an election. Recounting some counties but not others, they said, would be unfair to the voters whose ballots were not revisited. The proper business of the state is not to count the votes as accurately as possible—to know what actually happened—but to obey the formal protocol that tells us, in Hardy’s terms, who the winner should be defined to be.
More generally, formalism in the law manifests itself as an adherence to procedure and the words of the statutes, even when—or especially when—they cut against what common sense prescribes. Justice Antonin Scalia, the fiercest advocate of legal formalism there is, puts it very directly: “Long live formalism. It is what makes a government a government of laws and not of men.”
In Scalia’s view, when judges try to understand what the law intends—its spirit—they’re inevitably bamboozled by their own prejudices and desires. Better to stick to the words of the Constitution and the statutes, treating them as axioms from which judgments can be derived by something like logical deduction.
In questions of criminal justice, Scalia is equally devoted to formalism: the truth is, by definition, whatever a properly convened trial process determines it to be. Scalia makes this stance strikingly clear in his dissenting opinion in the 2009 case In Re Troy Anthony Davis, where he argued that a convicted murderer shouldn’t be granted a new evidentiary hearing, despite the fact that seven of the nine witnesses against him had recanted their testimony:
This Court has never held that the Constitution forbids the execution of a convicted defendant who has had a full and fair trial but is later able to convince a habeas court that he is “actually” innocent.
(Emphasis on “never” and scare quotes around “actually” both Scalia’s.)
As far as the court is concerned, Scalia says, what matters is the verdict arrived at by the jury. Davis was a murderer whether he killed anyone or not.
Chief Justice John Roberts isn’t a fervent advocate of formalism like Scalia, but he’s broadly in sympathy with Scalia’s philosophy. In his confirmation hearing in 2005, he famously described his job in
baseball terms:
Judges and justices are servants of the law, not the other way around. Judges are like umpires. Umpires don’t make the rules; they apply them. The role of an umpire and a judge is critical. They make sure everybody plays by the rules. But it is a limited role. Nobody ever went to a ball game to see the umpire.
Roberts, knowingly or not, was echoing Bill Klem, the “Old Arbitrator,” an umpire in the National League for almost forty years, who said, “The best-umpired game is the game in which the fans cannot recall the umpires who worked it.”
But the umpire’s role is not as limited as Roberts and Klem make it sound, because baseball is a formalist sport. To see this, you need look no further than game 1 of the 1996 American League Championship Series, in which the Baltimore Orioles faced the New York Yankees in the Bronx. Baltimore was leading in the bottom of the eighth when Yankee shortstop Derek Jeter launched a long fly ball to right field off Baltimore reliever Armando Benitez; well hit, but playable for center fielder Tony Tarasco, who settled under the ball and prepared to make the catch. That’s when twelve-year-old Yankee fan Jeffrey Maier, sitting in the front row of the bleachers, reached over the fence and pulled the ball into the stands.
Jeter knew it wasn’t a home run. Tarasco and Benitez knew it wasn’t a home run. Fifty-six thousand Yankee fans knew it wasn’t a home run. The only person in Yankee Stadium who didn’t see Maier reach over the fence was the only one who mattered, umpire Rich Garcia. Garcia called the ball a homer. Jeter didn’t try to correct the umpire’s call, much less refuse to jog around the bases and collect his game-tying run. No one would have expected that of him. That’s because baseball is a formalist sport. What a thing is is what an umpire declares it to be, and nothing else. Or, as Klem put it, in what must be the bluntest assertion of an ontological stance ever made by a professional sports official: “It ain’t nothin’ till I call it.”
How Not to Be Wrong : The Power of Mathematical Thinking (9780698163843) Page 39
