Read Is God a Mathematician? Online
Authors: Mario Livio
Is God a Mathematician? (10 page)
Another important discovery that Galileo made in 1610 was that of the phases of the planet Venus. In the geocentric doctrine, Venus
was assumed to move in a small circle (an
epicycle
) superimposed on its orbit around the Earth. The center of the epicycle was supposed to always lie on the line joining the Earth and the Sun (as in figure 17a; not drawn to scale). In this case, when observed from Earth, one would expect Venus always to appear as a crescent of somewhat varying width. In the Copernican system, on the other hand, Venus’s appearance should change from a small bright disk when the planet is on the other side of the Sun (as seen from Earth), to a large and almost dark disk when Venus is on the same side as Earth (figure 17b). Between those two positions Venus should pass through an entire sequence of phases similar to that of the Moon. Galileo corresponded about this important difference between the
predictions of the two doctrines with his former student Benedetto Castelli (1578–1643), and he conducted the crucial observations between October and December of 1610. The verdict was clear. The observations confirmed conclusively the Copernican prediction, proving that Venus indeed orbits the Sun. On December 11, a playful Galileo sent Kepler the obscure anagram “
Haec immatura a me iam frustra leguntur oy
” (“This was already tried by me in vain too early”). Kepler tried unsuccessfully to decipher the hidden message and eventually gave up. In his following letter, of January 1, 1611, Galileo finally transposed the letters in the anagram to read: “
Cynthiae figuras aemulatur mater amorum
” (“The mother of love [Venus] emulates the figures of Cynthia [the Moon]”).
Figure 17
All the findings I have described so far concerned either planets in the solar system—celestial bodies that orbit the Sun and reflect its light—or satellites revolving around these planets. Galileo also made two very significant discoveries related to stars—heavenly objects that generate their own light, such as the Sun. First, he performed observations of the Sun itself. In the Aristotelian worldview, the Sun was supposed to symbolize otherworldly perfection and immutability. Imagine the shock caused by the realization that the solar surface is far from perfect. It contains blemishes and dark spots that appear and disappear as the Sun rotates about its axis. Figure 18 shows Galileo’s hand-drawn images of sunspots, about which Galileo’s colleague Federico Cesi (1585–1630) wrote that they “delight both by the wonder of the spectacle and the accuracy of expression.” Actually, Galileo was neither the first to see sunspots nor even the first to write about them. One pamphlet in particular,
Three Letters on Sunspots,
written by the Jesuit priest and scientist Christopher Scheiner (1573–1650) annoyed Galileo so much that he felt compelled to publish an articulate reply. Scheiner argued that it was impossible for the spots to be right on the Sun’s surface. His claim was based partly on the spots being, in his opinion, too dark (he thought that they were darker than the dark parts of the Moon), and partly on the fact that they did not always appear to return to the same positions. Scheiner consequently believed that these were small planets orbiting the Sun. In his
History and Demonstrations Concerning Sunspots,
Galileo
systematically destroyed Scheiner’s arguments one by one. With a meticulousness, wit, and sarcasm that would have made Oscar Wilde jump to a standing ovation, Galileo showed that the spots were in fact not dark at all, only dark relative to the bright solar surface. In addition, Galileo’s work left no doubt that the spots were right on the Sun’s surface (I shall return to Galileo’s demonstration of this fact later in this chapter).
Figure 18
Galileo’s observations of other stars were truly the first human ventures into the cosmos that lies beyond our solar system. Unlike his experience with the Moon and the planets, Galileo discovered that his telescope hardly enlarged the images of stars at all. The implication was clear—stars were far more distant than planets. This was a surprise in itself, but what was truly eyepopping was the sheer
number
of new, faint stars that the telescope had revealed. In one small area around the constellation Orion alone, Galileo discovered no fewer than five hundred new stars. When Galileo turned his telescope to traverse the Milky Way—that patch of dim light that crosses the night
sky—he was in for the biggest surprise yet. Even the smooth-looking bright splash broke into a countless number of stars no human had ever seen as such before. The universe suddenly got much bigger. In the somewhat dispassionate language of a scientist, Galileo reported:
What was observed by us in the third place is the nature of matter of the Milky Way itself, which, with the aid of the spyglass, may be observed so well that all the disputes that for so many generations have vexed philosophers are destroyed by visible certainty, and we are liberated from worldly arguments. For the Galaxy is nothing else than a congeries of innumerable stars distributed in clusters. To whatever region of it you direct your spyglass, an immense number of stars immediately offer themselves to view. Of which very many appear rather large and very conspicuous but the multitude of small ones is truly unfathomable.
Some of Galileo’s contemporaries reacted enthusiastically. His discoveries ignited the imagination of scientists and non-scientists alike all over Europe. The Scottish poet Thomas Seggett raved:
Columbus gave man lands to conquer by bloodshed,
Galileo new worlds harmful to none.
Which is better?
Sir Henry Wotton, an English diplomat in Venice, managed to get hold of a copy of the
Sidereus Nuncius
the day that the book appeared. He immediately forwarded it to King James I of England, accompanied by a letter that read in part:
I send herewith unto his Majesty the strangest piece of news (as I may justly call it) that he hath ever yet received from my part of the world; which is the annexed book (come abroad this very day) of the Mathematical Professor of Padua, who by the help of an optical instrument…hath discovered four new planets rolling about the sphere of Jupiter, besides many other unknown fixed stars.
Entire volumes can be written (and indeed have been written) about all of Galileo’s achievements, but these lie beyond the scope of the present book. Here I only want to examine the effect that some of these astounding revelations had on Galileo’s views of the universe. In particular, what relation, if any, did he perceive between mathematics and the vast, unfolding cosmos?
The Grand Book of Nature
The philosopher of science Alexandre Koyré (1892–1964) remarked once that Galileo’s revolution in scientific thinking can be distilled to one essential element: the discovery that mathematics is the grammar of science. While the Aristotelians were happy with a qualitative description of nature, and even for that they appealed to Aristotle’s authority, Galileo insisted that scientists should listen to nature itself, and that the keys to deciphering the universe’s parlance were mathematical relations and geometrical models. The stark differences between the two approaches are exemplified by the writings of prominent members of the two camps. Here is the Aristotelian Giorgio Coresio: “Let us conclude, therefore, that he who does not want to work in darkness must consult Aristotle, the excellent interpreter of nature.” To which another Aristotelian, the Pisan philosopher Vincenzo di Grazia, adds:
Before we consider Galileo’s demonstrations, it seems necessary to prove how far from the truth are those who wish to prove natural facts by means of mathematical reasoning, among whom, if I am not mistaken, is Galileo. All the sciences and all the arts have their own principles and their own causes by means of which they demonstrate the special properties of their own object.
It follows that we are not allowed to use the principles of one science to prove the properties of another
[the emphasis is mine]. Therefore, anyone who thinks he can prove natural properties with mathematical argument is simply demented, for the two sciences are very different. The natural scientist studies natural bodies that have motion as
their natural and proper state, but the mathematician abstracts from all motion.
This idea of hermetic compartmentalization of the branches of science was precisely the type of notion that infuriated Galileo. In the draft of his treatise on hydrostatics,
Discourse on Floating Bodies,
he introduced mathematics as a powerful engine that enables humans to truly unravel nature’s secrets:
I expect a terrible rebuke from one of my adversaries, and I can almost hear him shouting in my ears that it is one thing to deal with matters physically and quite another to do so mathematically, and that geometers should stick to their fantasies, and not get involved in philosophical matters where the conclusions are different from those in mathematics. As if truth could ever be more than one; as if geometry in our day was an obstacle to the acquisition of true philosophy; as if it were impossible to be a geometer as well as a philosopher, so that we must infer as a necessary consequence that anyone who knows geometry cannot know physics, and cannot reason about and deal with physical matters physically! Consequences no less foolish than that of a certain physician who, moved by a fit of spleen, said that the great doctor Acquapendente [the Italian anatomist Hieronymus Fabricius (1537–1619) of Acquapendente], being a famous anatomist and surgeon, should content himself to remain among his scalpels and ointments without trying to effect cures by medicine, as if knowledge of surgery was opposed to medicine and destroyed it.
A simple example of how these different attitudes toward observational findings could completely alter the interpretation of natural phenomena is provided by the discovery of sunspots. As I noted earlier, the Jesuit astronomer Christopher Scheiner observed these spots competently and carefully. However, he made the mistake of allowing his Aristotelian prejudices of a perfect heaven to color his judgment. Consequently, when he discovered that the spots did not return to the
same position and order, he was quick to announce that he could “free the Sun from the injury of spots.” His premise of celestial immutability constrained his imagination and prevented him from considering the possibility that the spots could change, even beyond recognition. He therefore concluded that the spots
had
to be stars orbiting the Sun. Galileo’s course of attack on the question of the distance of the spots from the Sun’s surface was entirely different. He identified three observations that needed an explanation: First, the spots appeared to be thinner when they were near the edge of the solar disk than when they were near the disk’s center. Second, the separations between the spots appeared to increase as the spots approached the center of the disk. Finally, the spots appeared to travel faster near the center than close to the edge. Galileo was able to show with a single geometrical construction that the hypothesis—that the spots were contiguous to the surface of the Sun and were carried around by it—was consistent with all the observational facts. His detailed explanation was based on the visual phenomenon of
foreshortening
on a sphere—the fact that shapes appear thinner and closer together near the edge (figure 19 demonstrates the effect for circles drawn on a spherical surface).
The importance of Galileo’s demonstration for the foundations of the scientific process was tremendous. He showed that observational data become meaningful descriptions of reality only when embedded in an appropriate mathematical theory. The same observations could
lead to ambiguous interpretations unless understood in a broader theoretical context.
