Is God a Mathematician? (8 page)

In 1899, the Greek scholar A. Papadopoulos-Kerameus cataloged all the manuscripts that were housed in the Metochion, and the Archi
medes manuscript appeared as Ms. 355 on his list. Papadopoulos-Kerameus was able to read a few lines of the mathematical text, and perhaps realizing their potential importance, he printed those lines in his catalog. This was a turning point in the saga of this manuscript. The mathematical text in the catalog was brought to the attention of the Danish philologist Johan Ludvig Heiberg (1854–1928). Recognizing the text as belonging to Archimedes, Heiberg traveled to Istanbul in 1906, examined and photographed the palimpsest, and a year later announced his sensational discovery—two never-before-seen treatises of Archimedes and one previously known only from its Latin translation. Even though Heiberg was able to read and later publish parts of the manuscript in his book on Archimedes’ works, serious gaps remained. Unfortunately, sometime after 1908, the manuscript disappeared from Istanbul under mysterious circumstances, only to reappear in the possession of a Parisian family, who claimed to have had it since the 1920s. Improperly stored, the palimpsest had suffered some irreversible mold damage, and three pages previously transcribed by Heiberg were missing altogether. In addition, later than 1929 someone painted four Byzantine-style illuminations over four
pages. Eventually, the French family that held the manuscript sent it to Christie’s for auction. Ownership of the manuscript was disputed in federal court in New York in 1998. The Greek Orthodox Patriarchate of Jerusalem claimed that the manuscript had been stolen in the 1920s from one of its monasteries, but the judge ruled in favor of Christie’s. The palimpsest was subsequently auctioned at Christie’s on October 29, 1998, and it fetched $2 million from an anonymous buyer. The owner deposited the Archimedes manuscript at the Walters Art Museum in Baltimore, where it is still undergoing intensive conservation work and thorough examination. Modern imaging scientists have in their arsenal tools not available to the earlier researchers. Ultraviolet light, multispectral imaging, and even focused X-rays (to which the palimpsest was exposed at the Stanford Linear Accelerator Center) have already helped to decipher parts of the manuscript that had not been previously revealed. At the time of this writing, the careful scholarly study of the Archimedes manuscript is ongoing. I was fortunate enough to meet with the palimpsest’s forensic team, and figure 13 shows me next to the experimental setup as it illuminates one page of the palimpsest at different wavelengths.

Figure 12

The drama surrounding the palimpsest is only fitting for a document that gives us an unprecedented glimpse of the great geometer’s method.

The Method

When you read any book of Greek geometry, you cannot help but be impressed with the economy of style and the precision with which the theorems were stated and proved more than two millennia ago. What those books don’t normally do, however, is give you clear hints as to how those theorems were conceived in the first place. Archimedes’ exceptional document
The Method
partially fills in this intriguing gap—it reveals how Archimedes himself became convinced of the truth of certain theorems before he knew how to prove them. Here is part of what he wrote to the mathematician Eratosthenes of Cyrene (ca. 276–194 BC) in the introduction:

Figure 13

Archimedes touches here on one of the most important points in scientific and mathematical research—it is often more difficult to
discover what the important questions or theorems are than it is to find solutions to known questions or proofs to known theorems. So how did Archimedes discover some new theorems? Using his masterful understanding of mechanics, equilibrium, and the principles of the lever, he weighed in his mind solids or figures whose volumes or areas he was attempting to find against ones he already knew. After determining in this way the answer to the unknown area or volume, he found it much easier to prove geometrically the correctness of that answer. Consequently
The Method
starts with a number of statements on centers of gravity and only then proceeds to the geometrical propositions and their proofs.

Archimedes’ method is extraordinary in two respects. First, he has essentially introduced the concept of a
thought experiment
into rigorous research. The nineteenth century physicist Hans Christian Ørsted first dubbed this tool—an imaginary experiment conducted in lieu of a real one—
Gedankenexperiment
(in German: “an experiment conducted in the thought”). In physics, where this concept has been extremely fruitful, thought experiments are used either to provide insights prior to performing actual experiments or in cases where the real experiments cannot be carried out. Second, and more important, Archimedes freed mathematics from the somewhat artificial chains that Euclid and Plato had put on it. To these two individuals, there was one way, and one way only, to do mathematics. You had to start from the axioms and proceed by an inexorable sequence of logical steps, using well-prescribed tools. The free-spirited Archimedes, on the other hand, simply utilized every type of ammunition he could think of to formulate new problems and to solve them. He did not hesitate to explore and exploit the connections between the abstract mathematical objects (the Platonic forms) and physical reality (actual solids or flat objects) to advance his mathematics.

A final illustration that further solidifies Archimedes’ status as a magician is his anticipation of
integral and differential calculus
—a branch of mathematics formally developed by Newton (and independently by the German mathematician Leibniz) only at the end of the seventeenth century.

The basic idea behind the process of
integration
is quite simple
(once it is pointed out!). Suppose that you need to determine the area of the segment of an ellipse. You could divide the area into many rectangles of equal width and sum up the areas of those rectangles (figure 14). Clearly, the more rectangles you use, the closer the sum will get to the actual area of the segment. In other words, the area of the segment is really equal to the limit that the sum of rectangles approaches as the number of rectangles increases to infinity. Finding this limit is called
integration.
Archimedes used some version of the method I have just described to find the volumes and surface areas of the sphere, the cone, and of ellipsoids and paraboloids (the solids you get when you revolve ellipses or parabolas about their axes).

In
differential calculus,
one of the main goals is to find the slope of a straight line that is tangent to a curve at a given point, that is, the line that touches the curve only at that point. Archimedes solved this problem for the special case of a spiral, thereby peeping into the future work of Newton and Leibniz. Today, the areas of differential and integral calculus and their daughter branches form the basis on which most mathematical models are built, be it in physics, engineering, economics, or population dynamics.

Archimedes changed the world of mathematics and its perceived relation to the cosmos in a profound way. By displaying an astounding combination of theoretical and practical interests, he provided the first empirical, rather than mythical, evidence for an apparent mathematical design of nature. The perception of mathematics being the language of the universe, and therefore the concept of God as a mathematician, was born in Archimedes’ work. Still, there was something that Archimedes did not do—he never discussed the limitations of his
mathematical models when applied to actual physical circumstances. His theoretical discussions of levers, for instance, assumed that they were infinitely rigid and that rods had no weight. Consequently, he opened the door, to some extent, to the “saving the appearances” interpretation of mathematical models. This was the notion that mathematical models may only represent what is observed by humans, rather than describing the actual, true, physical reality. The Greek mathematician Geminus (ca. 10 BC–AD 60) was the first to discuss in some detail the difference between mathematical modeling and physical explanations in relation to the motion of celestial bodies. He distinguished between astronomers (or mathematicians), who, according to him, had only to suggest models that would
reproduce
the observed motions in the heavens, and physicists, who had to find
explanations
for the real motions. This particular distinction was going to come to a dramatic head at the time of Galileo, and I will return to it later in this chapter.

Figure 14

Somewhat surprisingly perhaps, Archimedes himself considered as one of his most cherished accomplishments the discovery that the volume of a sphere inscribed in a cylinder (figure 15) is always
²
/3 of the volume of the cylinder. He was so pleased with this result that he requested it be engraved on his tombstone. Some 137 years after Archimedes’ death, the famous Roman orator Marcus Tullius Cicero (ca. 106–43 BC) discovered the great mathematician’s grave. Here is Cicero’s rather moving description of the event:

Figure 15