After Fermat died, his son Samuel collected and published his writings, including the copy of Diophantus with all of Fermat’s marginal notes. In the 1700s, the Swiss mathematician Leonhard Euler took as a personal challenge to (re)-prove all of Fermat’s results in number theory. The statement about splitting up powers into like powers was the only one that eluded him. He did prove that the equations x3 + y3 = z3 and x4 + y4 = z4 have no integer solutions, but he despaired of finding a general method for all n.
Fermat’s innocent marginal note became known as “Fermat’s Last Theorem.” Technically, of course, it was not a theorem (i.e., a proven fact), but a conjecture. In 1825, Peter Gustav Lejeune Dirichlet proved that there are no whole-number solutions if n = 5. In 1839, Gabriel Lamé proved likewise for n = 7. By 1857, Ernst Kummer had proved it for all exponents n up to 100. Even though progress seemed agonizingly slow, the efforts to prove Fermat’s Last Theorem were opening up new areas of mathematics, today called algebraic number theory.
In the twentieth century, Fermat’s Last Theorem continued to spawn new mathematics, like the goose that laid golden eggs. In the early 1980s a German mathematician, Gerhard Frey, realized that any putative solution to Fermat’s equation, an + bn = cn, could be used to construct an auxiliary curve, given by the equation y2 = x(x – an)(x + bn), which struck Frey as a highly bizarre specimen. It was so bizarre, Frey argued, that it would violate another unproven conjecture in number theory, called the Taniyama–Shimura conjecture. The evidence was circumstantial at first, but subsequently an American mathematician, Kenneth Ribet, proved that Frey was right—if the Taniyama–Shimura conjecture was true, so was Fermat’s Last Theorem.
Frey’s idea was forehead-smackingly clever. He turned the variables in Fermat’s equation into coefficients of a different equation. It’s like the reversal of the foreground and background in a picture by M.C. Escher. Even so, it was very far from obvious that Frey’s and Ribet’s work represented any kind of breakthrough. They had only exchanged one seemingly unattainable goal for another. In effect, Frey and Ribet said: You want to climb Mount Everest? It’s easy. Just grow wings.
Only one person in the world actually believed that he could prove the Taniyama–Shimura conjecture: Andrew Wiles. And he did it more or less by “growing wings.” Actually, he built an airplane. Over a seven-year period, working alone in his attic, he linked together three of the most difficult, abstract, powerful theories of twentieth-century mathematics—the theories of L-functions, modular forms, and Galois representations—into a smoothly functioning machine. One might compare his proof to the Apollo missions to the Moon, which combined (at least) three independent technologies: rocketry, computing, and communications. None of these technologies were developed with a Moon mission in mind. If any one of the three had been missing, the Moon missions would have been inconceivable. Yet they did come together, at just the right time, to conquer a famous “unsolved problem” (How can humans fly to the Moon?). Coincidentally, like Fermat’s Last Theorem, that problem had been around for just about 350 years.
Above Woodcut by Maurits Cornelius Escher, 1938, Sky and Water I, an example of reversal of the foreground and background
WILES ANNOUNCED HIS PROOF of Fermat’s Last Theorem in 1993. Unlike Fermat, Wiles submitted his proof for publication, in 1994. In the three-and-a-half centuries between Fermat and Wiles, mathematicians had learned their lesson: a “theorem” without a published proof is no theorem at all. In fact, as Wiles wrote up his proof in 1993, he discovered a gap that took him a year (plus the assistance of a student, Richard Taylor) to plug. Perhaps, if Fermat had taken the trouble to write down his proof, he would have discovered a gap as well.
And this brings us to an inescapable question: Did Fermat actually find a correct proof? The answer of any competent number theorist would be a resounding no. According to André Weil, we can be certain that Fermat had a proof for the n = 4 case, and we may plausibly believe that he found something like Euler’s proof for the n = 3 case. Both of these cases were solvable with Fermat’s “technology.” But beginning with n = 5, the problem changes very significantly. The case n = 5 required the nineteenth-century machinery of complex numbers and algebraic number fields. And, as I have described, Wiles’ proof of the general case required top-of-the-line twentieth-century concepts that Fermat could never have dreamed of.
To the mathematical argument Weil adds a psychological one. Fermat repeatedly bragged about the n = 3 and n = 4 cases and posed them as challenges to other mathematicians (including poor Frénicle). But he never mentioned the general case, n = 5 and higher, in any of his letters. Why such restraint? Most likely, Weil argues, because Fermat had realized that his “truly wonderful proof” did not work in those cases. Every mathematician has had days like this. You think you have a great insight, but then you go out for a walk, or you come back to the problem the next day, and you realize that your great idea has a flaw. Sometimes you can go back and fix it. And sometimes you can’t.
Weil’s mathematical and psychological arguments are compelling. However, I would like to give the last word to a class of high-school students I taught in 1990, three years before Wiles announced his proof. On the last day of the course, a group of them performed a skit based on Fermat’s life. As the curtain came down, they chanted in unison:
“Fermat! Fermat! He’s our man! If he can’t prove it, no one can!”
10
an unexplored continent the fundamental theorem of calculus
William Dunham made an analogy between the discovery of Cardano’s formula for the cubic and Columbus’s discovery of America. However, the analogy falls short in one very important way. Columbus discovered an entire continent, comparable in size and importance to Europe. By contrast, Cardano’s formula today is little more than a curiosity, even to mathematicians. Perhaps its significance could be compared to the impact of Columbus’s discovery if Cuba and Hispaniola (where Columbus first made landfall) had merely been isolated islands with no continent nearby. It would certainly have been an amazing discovery, but perhaps not one to alter world history.
In the seventeenth century, though, mathematicians did find their equivalent of the New World, an unexplored “continent” of mathematics. The continent is called Calculus, and it had two primary discoverers: Isaac Newton and Gottfried Wilhelm Leibniz.
Calculus gave mathematicians and scientists a vocabulary for talking about quantities that change. The Fundamental Theorem provides a practical tool for solving problems about such quantities. Modern science, especially physics and engineering, would be inconceivable without it. Ironically, modern research mathematicians almost never use the term “calculus.” The branch of mathematics that deals with functions, integrals, derivatives, and infinite series—in other words, everything connected with the Fundamental Theorem of Calculus—is called “analysis,” and it is subdivided into Real Analysis, Complex Analysis, Functional Analysis, etc., just as America is subdivided into North, South, and Central America. To some extent the difference is one of intellectual rigor. In a “calculus” book, a mathematician will expect the arguments or explanations to be informal, intuitive, or entirely absent; in a book on “analysis,” he or she will expect formal and correct proofs. However, in my opinion, the distinction is also motivated (or perpetuated) by intellectual snobbery.
The functions f(x) and F(x) are continuous functions of a variable x. In the second equation, t is an auxiliary variable. F(x) is an antiderivative of f(x), meaning that dF/dx = f(x). The integral ∫ and derivative d/dx were operations introduced by Newton, based on the ancient problems of finding tangents to curves and areas of regions with curved edges. The Fundamental Theorem says that these are inverse operations: if you integrate the derivative of any function, or vice versa, you get back the original function.
Newton and Leibniz were born only four years apart—Newton in a village called Woolsthorpe, England, in 1642, and Leibniz in Leipzig, Germany, in 1646. Newton became a national hero in
England, and was buried in 1727 in Westminster Abbey, the final resting place of kings. Leibniz, in spite of his successes in both mathematics and philosophy, was relatively unappreciated in his home country. When he died in 1716, he was buried in an unmarked grave.
Newton made fundamental advances in physics as well as mathematics: He invented the reflecting telescope and formulated Newton’s laws of motion, which will be discussed at greater length in chapter 11. It is no exaggeration to say that our buildings stand and our spacecraft fly because of Newton’s laws. His long and productive scientific career more or less coincided with his years at Cambridge University, where he arrived in 1661 as a subsizar (undergraduate), and left in 1696 to manage the British mint.
Above Tangents and curves.
Leibniz, like Newton, had many interests outside mathematics. As a philosopher, he wrote (for example) about the problem of evil, and argued that although some evil was necessary, God had created the “best of all possible worlds.” This belief was later ridiculed in Voltaire’s famous book, Candide. Leibniz’s mathematical work was concentrated mostly between the years of 1672 and 1676, when he was stationed in Paris as a diplomat and had plenty of time on his hands. It was perhaps the best place in the world to learn mathematics, because much of Mersenne’s former network of friends was still intact, and had recently become formally organized as the French Academy of Sciences.
With the wisdom of hindsight, we can see that European mathematicians had been groping toward the discovery of calculus for the entire seventeenth century. Their attempts came from two separate directions. The first was the problem of quadrature, finding the areas of irregular (usually curved) regions. The problem of quadrature had fascinated mathematicians, of course, ever since antiquity. Early methods of computing areas were based on cut-and-rearrange arguments. Later, mathematicians like Archimedes in Greece and Liu Hui in China had become more sophisticated, approximating curved regions by a sequence of ever more accurate polygonal (straight-sided) regions.
In the early 1600s, an Italian mathematician, Buonaventura Cavalieri, had devised a systematic method, the “method of indivisibles,” that involves cutting the unknown area up into narrow, rectangular slices and adding up their areas. In fact Archimedes had developed a similar method centuries before, but his work had been lost and was recovered only in 1906, too late to materially influence the history of European science. Neither Cavalieri nor Archimedes understood how to turn this method into a practical calculation tool; the examples they worked out were few and arduous.
THE SECOND ROUTE TO CALCULUS began with the problem of drawing tangents to arbitrary curves. That is, how can you draw a line that just grazes a curve at one point? Like the quadrature problem, the solution in general requires a sequence of approximations. In order find the tangent line at a point on a curve, you need to know the slope of the curve at that point. To compute the slope, you can imagine taking a nearby point on the curve, drawing a line segment between the nearby point and the given point, and computing the slope of that line segment. Your answer will always be slightly off the mark. If only you could make the nearby point “infinitely close” and the line segment “infinitesimally short”! Unfortunately this is difficult to justify mathematically, because it amounts to dividing 0 by 0.
Some people, including Fermat and probably also Newton’s teacher at Cambridge, Isaac Barrow, had actually worked on both the problem of quadrature and the problem of tangents. However, only Newton and Leibniz grasped that the problems are actually flip sides of one another. If you have not studied calculus before, you should be shocked by this statement. There is absolutely no apparent connection between the tangent to a curve and the area inside a (different) curve.
The connection between the two ancient problems appears only after what seems at first like a highly arbitrary and artificial step: We turn curves into graphs. Of course, nowadays almost everybody is familiar with graphs. For example, there are graphs of stock prices in the business pages of the newspaper and electrocardiograms on monitors in hospitals. But in the seventeenth century, the idea of a graph was still very new.
A graph is a visual representation of the relationship between two variables: stock price and time, or electric potential and time. Some rule or process takes one variable (say, the time, or t) as input and produces the other variable (say, the stock price or f(t)) as output. The nature of this rule is not always clear in real-life examples like those that have been cited here. However, classical mathematicians were not interested in stock prices and electrocardiograms. They were interested in circles, parabolas, ellipses, spirals, and the like. For such curves, with a judicious choice of coordinate axes, it is often possible to write down a mathematical expression f(t) whose graph is the desired curve.
Below Color copper engraving of Leibniz (1646–1716), German philosopher and mathematician.
THE TWO ANCIENT PROBLEMS of tangency and quadrature are now more easily reinterpreted. The slope of a tangent line to a curve is actually a camouflaged version of the rate of change of the function it is a graph of. For example, suppose you went on a car trip, and at regular time intervals you recorded the distance on your odometer (let’s call that F(t)), while at the same time recording the speed on your speedometer (let’s call that f(t)). Thus, if you start at 12:00 and travel 60 miles in one hour, then F(1:00) = 60 miles. If you are going 30 mph at 1:00, then f(1:00) = 30 mph.
E49.05 GOTTFRIED WILHELM von LEIBNEZ (1646-1716). redit: The Granger Collection, New York
Now compare the odometer reading at time t to the next odometer reading, at time t’. These will give you two nearby points on the “odometer graph.” To find the slope of the graph of F(t), at time t, you would divide the distance you traveled in that short time interval by the elapsed time, in accordance with the high-school definition of slope as “rise over run.” But that is the same thing as computing the average speed! (For example, if you went 2 miles in 3 minutes, your average speed over that time was miles per minute, or 40 mph.) Thus, the rate of change of the “odometer function”—over a short time interval—is the average of the speedometer function over that length of time.
There is just one more change to make in order to express the relationship in calculus terms. The words “over a short time interval” must be erased and replaced with the word “instantaneous.” The instantaneous rate of change, or derivative, of the odometer function is the speedometer function. This seems like a tiny, almost insignificant modification. In fact, it is by far the hardest part of the whole argument. Neither Leibniz nor Newton fully justified it, and the debate over what exactly this word “instantaneous” means continued well into the nineteenth century.
However, let’s give Newton and Leibniz the benefit of the doubt and move on to the second classical problem, the problem of quadrature, this time focusing on the “speedometer function,” f(t). Again, there are two times, a and b (this time they don’t have to be close), and the aim is to discover the area (quadrature) of the region under the graph of f(t), and between the times a and b. Skipping the explanation and going straight to the answer (you can read the explanation in any book on calculus, if you’re brave), the quadrature of the “speedometer function” is the “odometer function.” In calculus lingo, F(t) is the integral of f(t).
THUS, NEWTON AND LEIBNIZ introduced two new mathematical concepts: the derivative, which solves the tangency problem, and the integral, which solves the quadrature problem (although Newton used different words for these.) To some extent, both of these things had been done before; the integral was essentially the same as Cavalieri’s method of indivisibles. But no one had realized before that the derivative and the interval are inverse operations. The derivative of the odometer function is the speedometer function; the integral of the speedometer function is the odometer function.
This inverse relationship is known today as the Fundamental Theorem of Calculus. Here is how we write it today as a formula (actually, it consists of two formulas):
> The first formula says that you can find the distance traveled, F(b) – F(a), by integrating the speed. (That is what the symbol ∫ means.) The second formula says that the speed is the rate of change, or derivative (that is what the symbol means), of the distance. Thus either one of the functions f(t) and F(t) can be calculated if you know the other.
It is as if one sailor sailed west from Europe to find China, and another sailed east from China to find Europe, and they met in the middle and shook hands at Panama. The first equation (metaphorically speaking) says that sailing west from Europe gets you to Panama, and the second one says that sailing east from China gets you to the same place.
WHY DID THIS DISCOVERY open up a new continent of mathematics? It finally gave mathematicians absolute control over the concept of continuous change. Remember that continuous motion had puzzled the Greeks ever since Zeno put forth his famous paradoxes. Before Leibniz and Newton, mathematicians had been limited to static diagrams or discrete quantities. The world of continuous motion and continuously varying quantities was closed to them. But modern science is all about change. In calculus, mathematicians found the necessary vocabulary to do modern science.
Right Illustration from The method of fluxions and infinite series by Isaac Newton (1642–1727), first published in London in 1736.
Calculus is an immensely practical tool. Before Newton and Leibniz, finding an area or computing a slope was an unbelievably laborious process. But one of the benefits of having two routes to Panama (to continue the analogy from above) is that you can pick the route that is more convenient. Very often, one route will turn out to be much more convenient.
With calculus, a mathematics student can now, in an afternoon, compute a better approximation of pi than Archimedes or Liu Hui. In fact, both Newton and Leibniz delighted in finding new expressions for pi and other constants. Tables of logarithms and sines—indispensible to mathematicians, engineers, and astronomers in the pre-computer age—could be computed routinely, and as precisely as patience allowed. Volumes, areas, and lengths of curves that had taken mathematicians centuries to figure out were now computable. Even as recently as the 1630s, Descartes had written that it was impossible to rectify a curve—that is, to find a straight line of equal length. With calculus, even a student can do it.†
The Universe in Zero Words Page 7