Sunday, June 3, 2007

Kepler's Most Beautiful Idea

Johannes Kepler, the 17th-century astronomer credited with discovering that the planets’ paths around the sun are ellipses, was teaching geometry to a bunch of bored kids in Graz, Austria, when he was suddenly struck by the most beautiful idea he would ever have.
Six planets were known at the time --Mercury, Venus, Earth, Mars, Jupiter, and Saturn. Their orbits, which Kepler still believed were circles, placed them at certain distances from the sun. Why six planets? Why those particular distances? Kepler wondered. In the true spirit of modern science, he was confident that the distances of the planets were governed by some sort of mathematical law.
It is a well known geometrical fact that if you wish to construct solid, three-dimensional bodies whose sides are regular polygons (polygons whose sides are all the same length), try as you might, you will end up with no more than five such bodies. This result is a geometrical theorem --in 3D space there exist five and only five regular, or “Pythagorean,” solids. It has nothing to do with the state of our mathematical knowledge or our technological prowess. Like the statement that parallel lines meet at infinity, the existence of no more than five regular solids is an inescapable property of three-dimensional flat space.
Kepler’s idea was this: there are only six planets because there are only five regular solids and if we put these solids one inside the other in a nested pattern, the spheres defining the boundaries where an inner solid touches the next one out have radii which are in the same proportion as the distances of the planets. Kepler called this hypothesis the Mysterium Cosmographicum . It was beautiful in its geometrical simplicity and awe-inspiring in its depth of implication (the existence of god the mathematician). It rang true. But it was not.
Kepler tried hard to make the nested regular solids match the distances of the planets, but to no avail. He had based his calculations on figures obtained by Copernicus seventy years earlier. Perhaps more accurate measurements of the distances might show them to fit his elegant scheme?
The greatest observational astronomer of the age, Tycho Brahe, had performed such measurements. When Kepler got hold of the data after Tycho’s death he was disappointed. In the end --and in view of Galileo’s subsequent discovery of Jupiter’s four large moons, which the Mysterium could not possibly accomodate-- Kepler reluctantly abandoned his pet theory. It took courage, but Kepler was imbued with “the will to find out” as opposed to “the will to believe”. He was skeptical even of his own ideas.
Skepticism is essential in science. It prevents us from foisting our emotions on the universe, and thus helps us to study verifiable facts with a cool head. Gullibility readily accepts claims that may seem satisfying on grounds of personal taste or convenience, but that all too often are simply not true. The Argentinean philosopher of science Mario Bunge wrote:
Scientific knowledge is sometimes unpleasant. It often contradicts the classics; occasionally tortures common sense and humiliates intuition. Lastly, it may prove convenient for some, but not for others. The hallmark of scientific knowledge is that it is verifiable.
Personal taste, appeals to authority, and even democracy (like in deciding by popular vote which one of several contending hypotheses is true) have no place in science. Science is about discovering objective facts whose truth does not depend on who champions or opposes them. As the French mathematician Henri Poincaré said: “The sole source of truth is experiment. Only it can teach us something new; only it can give us certainty.”

Thursday, May 31, 2007

The Nurturing Sky

Few people nowadays know their way around the starry dome, and it is a rare city-dweller who can identify a constellation other than the Big Dipper. Our forebears, though, had good reason to pay close attention to celestial occurrences. All too often their livelihoods --and their very lives-- depended on correctly interpreting what they saw in the sky.
Ancient Egyptians noticed that the annual flooding of the Nile River took place at the same time of year when the star Sirius first appeared in the dawn sky. Native Americans also read the course of the seasons in the sky. “The old men used to study the stars carefully and in this way could tell when each season began,” a Cahuila Indian told a researcher in the 1920s.

"They would meet in the ceremonial house and argue about the time certain stars would appear and would often gamble about it. This was a very important matter, for upon the appearance of certain stars depended the season of crops... They never went to the mountains until they saw a certain star, for they knew they could not find food there previously."

Early navigators from around the world steered their vessels by the map provided by the constellations. The Chinese called their see-going barges “starry rafts,” and slaves escaping north through Georgia and Mississippi in the days before the American Civil War were advised to “follow the drinking gourd,” that is, the Big Dipper.
Celestial phenomena were long held to be omens of great misfortune to come. In Hamlet, Shakespeare has Horatio discourse on the portents to be read in the sky:

In the most high and balmy state of Rome
A little ere the mightiest Julius fell
The graves stood tenantless and the sheeted dead
Did squeak and gibber in the Roman streets
As, stars with trains of fire, and dews of blood,
Disasters in the sun; and the moist star...
Was sick almost to doomsday with eclipse.

The Franciscan friar Bernardino de Sahagún, author of Historia general de las cosas de la Nueva España , wrote of the Aztecs: “They called comets citlalin popoca, which means smoking star. They believed comets presaged the death of a prince or king, or the onset of war and famine.”
All things considered, it is easy to see why all ancient civilizations linked the stars and their movements to their own destinies, and why they invented gods and heroes to people the starry dome. What is not so easy to understand is why some people today still believe in astral influence on human affairs. Through the ages we have studied the stars, we have gazed at them in wonder and awe, and we have on occasion foisted our dreams and wishes on them. But when we finally scrutinized them with a good dose of healthy skepticism, probing them with scientifically oriented minds, we found their inner workings to be much more wonderful and awe-inspiring than anything the astrologers ever dreamed of.
Early observations of the starry sky and primitive superstitions were a long and necessary prelude to modern science. The doctrines of pseudoscience --astrology, UFOs and the like-- pale in comparison with even the simplest findings of science. As Carl Sagan wrote: “Science is more intricate and subtle, reveals a much richer universe, and powerfully evokes our sense of wonder. And it has the additional and important virtue --to whatever extent the term has any meaning-- of being true.”

Einstein and God

Albert Einstein was a deeply religious man. But, like Baruch Spinoza before him, he understood religion in a very idiosyncratic way. A Jew by birth, he very early abandoned traditional religious views.
On April 24, 1929, Einstein, who by then was the most famous scientist ever, received a simple cablegram from Rabbi Herbert Goldstein of the Institutional Synagogue, New York: “Do you believe in God?” Einstein replied: “I believe in Spinoza’s God who reveals himself in the orderly harmony of what exists, not in a God who concerns himself with fates and actions of human beings.”
Einstein’s major utterances on the subject of religion came in the form of two essays with mirror-image titles --”Religion and Science,” written for the New York Times on the occasion of his second visit to the United States in 1931, and “Science and Religion,” written eight years later. In the first of these Einstein outlines three states of religious development --the religion of fear that moved primitive peoples, the subsequent moral religion whose driving force is social feeling, and finally, the “cosmic religious sense...which recognizes neither dogmas nor God made in man’s image.”
People tend to equate “religion” with their particular system of belief. In his second essay Einstein was far more ecumenical. To him religion stems from the legitimate need to find a goal for our human aspirations. The purely rational --or scientific-- conception of existence cannot provide an ethics to guide human endeavor; it cannot give us a sense of the ultimate and fundamental ends.
"To make clear these fundamental ends and valuations, and to set them fast in the emotional life of the individual, seems to me precisely the most important function which religion has to perform in the social life of man. And if one asks whence derives the authority of such fundamental ends, since they cannot be stated and justified merely by reason, one can only answer: they exist in a healthy society as powerful traditions..."
At a loss to describe succintly what religion is, Einstein is content to ask what characterizes a religious person.
"A person who is religiously enlightened appears to me to be one who has...liberated himself from the fetters of his selfish desires and is preoccupied with thoughts, feelings, and aspirations to which he clings because of their super-personal value. It seems to me that what is important is the force of this super-personal content and the depth of the conviction concerning its overpowering meaningfulness, regardless of whether any attempt is made to unite this content with a divine Being."
Einstein describes as religious the scientist’s trust in the rational nature of reality. But one must not confuse the religious with the supernatural. “The more a man is imbued with the ordered regularity of all events,” he writes, “the firmer becomes his conviction that there is no room left by the side of this ordered regularity for causes of a different nature.”
The awe that deep scientific understanding brings is very close to a religious epiphany and stems from the same underlying psychological foundation.
"Whoever has undergone the intense experience of [scientific understanding] is moved by profound reverence for the rationality made manifest in existence. By way of the understanding he achieves a far reaching emancipation from the shackles of personal hopes and desires, and thereby attains that humble attitude of mind towards the grandeur of reason incarnate in existence... . This attitude...appears to me to be religious in the highest sense of the word."
I wrote this entry in response to an e-mail from my good friend Dimitri Maekawa, who is experiencing an Einsteinian epiphany in Victoria, British Columbia. May he find solace in communion with such a one as uncle Albert.