Descartes (2)

 

In Rules for the Direction of the Mind, Descartes wrote: “The first principles themselves are given by intuition alone, while, on the contrary, the remote conclusions are furnished only by deduction. . . . These two methods are the most certain routes to knowledge, and the mind should admit no others. All the rest should be rejected as suspect of error and dangerous.” 

Descartes was embracing the Euclidean ideal: Start from self-evident axioms, proceed by infallible deductions. But in his own research, Descartes forgot the Euclidean ideal. Nowhere in the Geometry do we find the label Axiom, Theorem, or Proof. 

In classical Greece, and again in the Renaissance and after, mathematicians distinguished two ways of proceeding—the “synthetic” and the “analytic.” The synthetic way was Euclid’s: from axioms through deductions to theorems. In the analytic mode, you start with a problem and “analyze” it to find a solution. Today we might call this a “heuristic” or “problem-solving” approach. 

In formal presentation of academic mathematics, the synthetic was and still is the norm. Foundationist schools of the nineteenth and twentieth centuries identify mathematics with its synthetic mode—true axioms followed by correct deductions to yield guaranteed true conclusions. 

In his Rules for the Direction of the Mind, Descartes insists on the synthetic method. But his own research, in the Geometry, uses only the analytic mode. He solves problems. He finds efficient methods for solving problems. Never does he bother with axioms. 

Descartes’s conviction of the certainty of mathematics might lead readers to expect that at least Descartes’s own mathematics is error-free. But of course, as we will see, the Geometry, like every other math book, has mistakes. Certitude is only a goal.

**

You won’t find in the Geometry the method we teach nowadays as Cartesian or “analytic” geometry. Our analytic geometry is based on rectangular coordinates (which we call “Cartesian”). To every point in the plane we associate a pair of real numbers, the “x” and “y” coordinates of the point. To an equation relating x and y corresponds a “graph”—the set of points whose x and y coordinates satisfy the equation. For an equation of first degree, the graph is a straight line. For an equation of second degree, it’s a circle or other conic section. Our idea is to solve geometric problems by reducing them to algebra. Nowhere in Descartes’s book do we see these familiar horizontal and vertical axes! Boyer says it was Newton who first used orthogonal coordinate axes in analytic geometry.

**
The conceptual essence of analytic geometry, the “isomorphism” or exact translation between algebra and geometry, was understood more clearly by Fermat than by Descartes. Fermat’s analytic geometry predated Descartes’s, but it wasn’t published until 1679. The modern formulation comes from a long development. Fermat and Descartes were the first steps. Instead of systematically developing the technique of orthogonal coordinate axes, the Geometry studies a group of problems centering around a problem of Pappus of Alexandria (third century A.D.). To solve Pappus’s problem Descartes develops an algebraic-geometric procedure. First he derives an algebraic equation relating known and unknown lengths in the problem. But he doesn’t then look for an algebraic or numerical solution, as we would do. He is faithful to the Greek conception, that by a solution to a geometric problem is meant a construction with specified instruments. When possible Descartes uses the Euclidean straight edge and compass. When necessary, he brings in his own instrument, an apparatus of hinged rulers. Algebra is an intermediate device, in going from geometric problem to geometric solution. Its role is to reduce a complicated curve to a simpler one whose construction is known. He solves third and fourth-degree equations by reducing them to second degree—to conic sections. He solves certain fifth-and sixth-degree equations by reducing them to third degree. A modern reader knows that the general equation of fifth degree can’t be solved by extraction of roots. So he’s skeptical about Descartes’s claim that his hinged rulers can solve equations of degree six and higher. Descartes was mistaken on several points. In themselves, these are of little interest today. But they discredit his claim of absolute certainty. Descartes’s mathematics refutes his epistemology. 

Emily Grosholz and Carl Boyer point out errors in the Geometry. “When he turns his attention to the locus of five lines, he considers only a few cases, not bothering to complete the task, because, as he says, his method furnishes a way to describe them. But Descartes could not have completed the task, which amounted to giving a catalogue of the cubics. . . . Newton, because he was able to move with confidence between graph and equation, first attempted a catalogue of the cubics; he distinguished seventy-two species of cubics, and even then omitted six” (Grosholz, referring to Whiteside).

**

Descartes claimed his Method was infallible in science and mathematics. He was more cautious with religion. He didn’t derive Holy Scripture or divine revelation by self-evident axioms and infallible deductions. When he heard that Galileo’s Dialogue on the Two Chief Systems was condemned by the Holy Church, he suppressed his first book, Le Monde, even though he was living in Holland, safe from the Church. (Galileo was kept under house arrest at first. For three years he had to recite the seven penitential psalms every week.) Descartes wrote to Father Mersenne, “I would not want for anything in the world to be the author of a work where there was the slightest word of which the Church might disapprove.”

**

Like Pascal, Newton, and Leibniz, Descartes may have valued his contributions to theology above his mathematics. His struggle against skeptics and heretics is the major half of his philosophy, more explicit than his battles with scholastics. “In Descartes’ reply to the objections of Father Bourdin, he announced that he was the first of all men to overthrow the doubts of the Sceptics . . . he discovered how the best minds of the day either spent their time advocating scepticism, or accepted only probable and possibly uncertain views, instead of seeking absolute truth. . . . It was in the light of this awakening to the sceptical menace, that when he was in Paris Descartes set in motion his philosophical revolution by discovering something so certain and so assured that all the most extravagant suppositions brought forward by the sceptics were incapable of shaking . . . in the tradition of the greatest medieval minds, (he) sought to secure man’s natural knowledge to the strongest possible foundation, the all-powerful eternal God” (Popkin, p. 72). The essence of the Meditations is a proof that the world exists by first proving Descartes exists, and then, by contemplating Descartes’s thoughts, proving that God exists and is not a deceiver. Once a non-deceiving God exists, everything else is easy.

 

How Not To Be Wrong

So here’s the question. You don’t want your planes to get shot down by enemy fighters, so you armor them. But armor makes the plane heavier, and heavier planes are less maneuverable and use more fuel. Armoring the planes too much is a problem; armoring the planes too little is a problem.

Somewhere in between there’s an optimum. The reason you have a team of mathematicians socked away in an apartment in New York City is to figure out where that optimum is.

The military came to the SRG with some data they thought might be useful. When American planes came back from engagements over Europe, they were covered in bullet holes. But the damage wasn’t uniformly distributed across the aircraft. There were more bullet holes in the fuselage, not so many in the engines.

The officers saw an opportunity for efficiency; you can get the same protection with less armor if you concentrate the armor on the places with the greatest need, where the planes are getting hit the most. But exactly how much more armor belonged on those parts of the plane? That was the answer they came to Wald for. It wasn’t the answer they got.

The armor, said Wald, doesn’t go where the bullet holes are. It goes where the bullet holes aren’t: on the engines.

Wald’s insight was simply to ask: where are the missing holes? The ones that would have been all over the engine casing, if the damage had been spread equally all over the plane? Wald was pretty sure he knew. The missing bullet holes were on the missing planes. The reason planes were coming back with fewer hits to the engine is that planes that got hit in the engine weren’t coming back. Whereas the large number of planes returning to base with a thoroughly Swiss-cheesed fuselage is pretty strong evidence that hits to the fuselage can (and therefore should) be tolerated. If you go the recovery room at the hospital, you’ll see a lot more people with bullet holes in their legs than people with bullet holes in their chests. But that’s not because people don’t get shot in the chest; it’s because the people who get shot in the chest don’t recover.

Here’s an old mathematician’s trick that makes the picture perfectly clear: set some variables to zero. In this case, the variable to tweak is the probability that a plane that takes a hit to the engine manages to stay in the air. Setting that probability to zero means a single shot to the engine is guaranteed to bring the plane down. What would the data look like then? You’d have planes coming back with bullet holes all over the wings, the fuselage, the nose—but none at all on the engine. The military analyst has two options for explaining this: either the German bullets just happen to hit every part of the plane but one, or the engine is a point of total vulnerability. Both stories explain the data, but the latter makes a lot more sense. The armor goes where the bullet holes aren’t.

Wald’s recommendations were quickly put into effect, and were still being used by the navy and the air force through the wars in Korea and Vietnam. I can’t tell you exactly how many American planes they saved, though the data-slinging descendants of the SRG inside today’s military no doubt have a pretty good idea. One thing the American defense establishment has traditionally understood very well is that countries don’t win wars just by being braver than the other side, or freer, or slightly preferred by God. The winners are usually the guys who get 5% fewer of their planes shot down, or use 5% less fuel, or get 5% more nutrition into their infantry at 95% of the cost. That’s not the stuff war movies are made of, but it’s the stuff wars are made of. And there’s math every step of the way.

**

If your acquaintance with mathematics comes entirely from school, you have been told a story that is very limited, and in some important ways false. School mathematics is largely made up of a sequence of facts and rules, facts which are certain, rules which come from a higher authority and cannot be questioned. It treats mathematical matters as completely settled.

Mathematics is not settled. Even concerning the basic objects of study, like numbers and geometric figures, our ignorance is much greater than our knowledge. And the things we do know were arrived at only after massive effort, contention, and confusion. All this sweat and tumult is carefully screened off in your textbook.

**

Mathematical facts can be simple or complicated, and they can be shallow or profound. This divides the mathematical universe into four quadrants:

“Basic arithmetic facts, like 1 + 2 = 3, are simple and shallow. So are basic identities like sin(2x) = 2 sin x cos x or the quadratic formula: they might be slightly harder to convince yourself of than 1 + 2 = 3, but in the end they don’t have much conceptual heft.

Moving over to complicated/shallow, you have the problem of multiplying two ten-digit numbers, or the computation of an intricate definite integral, or, given a couple of years of graduate school, the trace of Frobenius on a modular form of conductor 2377. It’s conceivable you might, for some reason, need to know the answer to such a problem, and it’s undeniable that it would be somewhere between annoying and impossible to work it out by hand; or, as in the case of the modular form, it might take some serious schooling even to understand what’s being asked for. But knowing those answers doesn’t really enrich your knowledge about the world.

The complicated/profound quadrant is where professional mathematicians like me try to spend most of our time. That’s where the celebrity theorems and conjectures live: the Riemann Hypothesis, Fermat’s Last Theorem,* the Poincaré Conjecture, P vs. NP, Gödel’s Theorem . . . Each one of these theorems involves ideas of deep meaning, fundamental importance, mind-blowing beauty, and brutal technicality, and each of them is the protagonist of books of its own.

But not this book. This book is going to hang out in the upper left quadrant: simple and profound. The mathematical ideas we want to address are ones that can be engaged with directly and profitably, whether your mathematical training stops at pre-algebra or extends much further. And they are not “mere facts,” like a simple statement of arithmetic—they are principles, whose application extends far beyond the things you’re used to thinking of as mathematical. They are the go-to tools on the utility belt, and used properly they will help you not be wrong.

**

The more abstract and distant from lived experience my research got, the more I started to notice how much math was going on in the world outside the walls. Not Galois representations or cohomology, but ideas that were simpler, older, and just as deep—the northwest quadrant of the conceptual foursquare. I started writing articles for magazines and newspapers about the way the world looked through a mathematical lens, and I found, to my surprise, that even people who said they hated math were willing to read them. It was a kind of math teaching, but very different from what we do in a classroom.

Algebra

Algebra is the generic term for the maths of equations, in which numbers and operations are written as symbols. The word itself has a curious history. In medieval Spain, barbershops displayed signs saying Algebrista y Sangrador. The phrase means ‘Bonesetter and Bloodletter’, two trades that used to be part of a barber’s repertoire. (This is why a barber’s pole has red and white stripes—the red symbolizes blood, and the white symbolizes the bandage.)

The root of algebrista is the Arabic al-jabr, which, in addition to referring to crude surgical techniques, also means restoration or reunion. In ninth-century Baghdad, Muhammad ibn Musa al-Khwarizmi wrote a maths primer entitled Hisab al-jabr w’al-muqabala, or Calculation by Restoration and Reduction.

**

Al-Khwarizmi wasn’t the first person to use restoration and reduction—these operations could also be found in Diophantus; but when Al-Khwarizmi’s book was translated into Latin, the al-jabr in the title became algebra. Al-Khwarizmi’s algebra book, together with another one he wrote on the Indian decimal system, became so widespread in Europe that his name was immortalized as a scientific term: Al-Khwarizmi became Alchoarismi, Algorismi and, eventually, algorithm.

**

Between the fifteenth and seventeenth centuries mathematical sentences moved from rhetorical to symbolic expression. Slowly, words were replaced with letters. Diophantus might have started letter symbolism with his introduction of  for the unknown quantity, but the first person to effectively popularize the habit was François Viète in sixteenth-century France. Viète suggested that upper-case vowels—A, E, I, O, U—and Y be used for unknown quantities, and that the consonants B, C, D, etc., be used for known quantities.

Within a few decades of Viète’s death, René Descartes published his Discourse on Method. In it, he applied mathematical reasoning to human thought. He started by doubting all of his beliefs and, after stripping everything away, was left with only certainty that he existed. The argument that one cannot doubt one’s own existence, since the process of thinking requires the existence of a thinker, was summed up in the Discourse as I think, therefore I am. The statement is one of the most famous quotations of all time, and the book is considered a cornerstone of Western philosophy. Descartes had originally intended it as an introduction to three appendices of his other scientific works. One of them, La Géométrie, was equally a landmark in the history of maths.

In La Géométrie Descartes introduces what has become standard algebraic notation. It is the first book that looks like a modern maths book, full of as, bs and cs and xs, ys and zs. It was Descartes’s decision to use lower-case letters from the beginning of the alphabet for known quantities, and lower-case letters from the end of the alphabet for the unknowns. When the book was being printed, however, the printer started to run out of letters. He enquired if it mattered if x, y or z was used. Descartes replied not, so the printer chose to concentrate on x since it is used less frequently in French than y or z. As a result, x became fixed in maths—and the wider culture—as the symbol for the unknown quantity. That is why paranormal happenings are classified in the X-Files and why Wilhelm Röntgen came up with the term X-ray. Were it not for issues of limited printing stock, the Y-factor could have become a phrase to describe intangible star quality and the African-American political leader might have gone by the name Malcolm Z.

With Descartes’ symbology, all traces of rhetorical expression had been expunged.

**

In 1621, a Latin translation of Diophantus’s masterpiece Arithmetica was published in France. The new edition rekindled interest in ancient problem-solving techniques, which, combined with better numerical and symbolic notation, ushered in a new era of mathematical thought. Less convoluted notation allowed greater clarity in describing problems. Pierre de Fermat, a civil servant and judge living in Toulouse, was an enthusiastic amateur mathematician who filled his own copy of Arithmetica with numerical musings. Next to a section dealing with Pythagorean triples—any set of natural numbers a, b and c such that a²+ b² = c², for example 3, 4 and 5—Fermat scribbled some notes in the margin. He had noticed that it was impossible to find values for a, b and c such that a³ + b³= c³. He was also unable to find values for a, b and c such that 

a⁴+ b⁴ = c⁴. Fermat wrote in his Arithmetica that for any number n greater than 2, there were no possible values a, b and c that satisfied the equation aⁿ + bⁿ = cⁿ. ‘I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain,’ he wrote.

Fermat never produced a proof—marvellous or otherwise—of his proposition even when unconstrained by narrow margins. His jottings in Arithmetica may have been an indication that he had a proof, or he may have believed he had a proof, or he may have been trying to be provocative. In any case, his cheeky sentence was fantastic bait to generations of mathematicians. The proposition became known as Fermat’s Last Theorem and was the most famous unsolved problem in maths until the Briton Andrew Wiles cracked it in 1995. Algebra can be very humbling in this way—ease in stating a problem has no correlation with ease in solving it. Wiles’s proof is so complicated that it is probably understood by no more than a couple of hundred people.