Mathematics and the Real World (15 page)

Hipparchus, and after him Ptolemy, adopted the geometric principles of Apollonius. Ptolemy was active in Alexandria from the year 90 CE to 169 CE, but these are approximations, as no biographies with details of his life
have survived. His writings, however, have been preserved almost in their entirety, as they were copied and kept by the Arabs, and they provide much information on the model of the celestial bodies and Ptolemy's scientific approach. His main contribution was technical rather than conceptual. His basic starting point was the assumption that the heavenly bodies orbited the Earth, that their paths were epicycles whose centers were on a deferent that revolved around a point in space that was not necessarily the center of the Earth, and that the planes on which those deferents revolved were probably angled toward each other. With intensive and detailed computations, Ptolemy constructed a model, complete for his time, of the motion of the heavenly bodies. The model included seventy-two epicycles and larger cycles and represented a marked improvement on earlier models as it provided a superior fit of the observations to the model. The model gave far more accurate predictions than did the previous ones, and it remained the main model of the motion of celestial bodies for fifteen hundred years, longer than any other physical model in the history of science.

Ptolemy was aware of other ideas and approaches put forward by the Greek scientists over time and actually referred to them in his writings. He rejected Aristarchus's idea that the Earth revolved around the Sun with a quite rational claim based on calculations. Ptolemy's calculations showed that according to the heliocentric method with the Earth orbiting the Sun, the speed of the motion is incomprehensible and human thought cannot even imagine such a high speed. In particular he claimed that if the Earth revolved around the Sun, all animals and humans would fall off. He rejected with a similar argument the view still held at the time by Heraclides, namely, that the Earth revolved on its own axis. He added another argument to this, stating that if the Earth revolved on its own axis, a stone thrown upward would not fall straight down. He answered Heraclides's claim that huge cosmological bodies could not revolve around the relatively small Earth with the retort that the size of the bodies was not the determining factor, but their weight, and the Earth was very heavy, while the weight of the sky, the stars, and the ether in which they are situated is negligible. These are completely rational claims based on what our senses show us. Ptolemy acknowledged that Aristarchus's model, with the Earth
revolving around the Sun, was simpler. He knew that there was contradiction between the simplicity of the mathematical theory on the one hand and what our senses perceive and the precision of the results on the other. In this situation, Plato would have preferred the simple mathematical theory. But Ptolemy's mathematical theory was so exact that it superseded the simplicity of other notions. This contrast between what our senses tell us and a simple but abstract model has accompanied science throughout the generations.

It should be stated that Ptolemy repeatedly said in his books that his model is just a mathematical description of nature. Specifically, he did not claim that his circles were a law of nature but that it was mathematics that best described nature. Moreover, he took the trouble to show that in some cases simpler models could yield more accurate results, but he was searching for a mathematical model that would incorporate all the phenomena. It was the Christian church that adopted Ptolemy's model as scientific truth and declared that God used mathematics to create the world.

What causes the tides? • Why did Descartes say “I think, therefore I am”? • What is the Harmony of the World? • What effect does a new star have? • How can the third derivative help politicians get elected? • Why don't you need to be afraid of differential equations? • Why does the wall try to push back at us? • How was the mystery of the relation between the length of the string and the note it produces solved
?

15. THE SUN REVERTS TO THE CENTER

The Renaissance period that started in the fifteenth century, also known as the early modern period, brought far-reaching developments in society, culture, and politics. These developments also encompassed a scientific revolution whose principles completely changed science and are as relevant today as they were then. Mathematics played a major role in that revolution.

Ptolemy's model of the heavenly bodies did not change much from when he formulated it in the second century CE until the sixteenth century. The model served as a sufficiently precise tool for predicting astronomical events, drawing up calendars, and so on. Ptolemy's model and the mathematics it was based on were studied in colleges and universities throughout the Middle East and Europe. The Arabs, who were at the forefront of scientific development at that time, enhanced the accuracy of the model by
adding epicycles. At its peak the model incorporated seventy-seven deferents and epicycles along which the heavenly bodies moved in their orbits around the Earth. The complexity of the model, however, although it resulted in greater accuracy, also undermined Ptolemy's model.

Nicolaus Copernicus was born in 1473 in Thorn, Prussia, then part of the Great Polish Kingdom. He studied first in Cracow, which was a famous center of science, but most of his advanced studies he completed in Italy. There he became familiar with the Greek scientific literature, both the more classical and the writings of Ptolemy. He was a polymath. He knew several languages, including Latin and Greek, completed the study of law and medicine, and actually practiced medicine—all while studying mathematics and science, as well as astrology, then a highly respected profession. He then returned to Prussia and served as secretary, doctor, and astrologer to the bishop of Warmia, and then as economic administrator and advisor to the Warmia parliament. Copernicus devoted much time to astronomy, but it was not his sole occupation.

The idea of adopting and enhancing Aristarchus's heliocentric model came to him in the course of his studies in Italy. In his writings Copernicus refers to the effect of Aristarchus's model and the earlier ideas of the Pythagoreans. As early as in 1510 he wrote an essay on the principles of the model in which the Sun is at the center of the universe and the heavens and the planets, including the Earth, revolve around it. However, he circulated the essay to only a few colleagues. In the next few years he continued with the mathematical development of the model, including the completion of the astronomical measurements that he carried out himself over many years. His work was almost completed in 1533, but Copernicus still did not publish it. Nevertheless, news of it reached Europe, and requests for copies and encouragement for him to complete his work arrived from all over the continent. Scientists lectured widely on his theory, including in Rome before Pope Clement VII and some of his cardinals. They showed great interest in the findings and asked for copies of the work and the accompanying astronomical tables.

The Church, in general, did not object to the use of mathematics to
describe nature and justified it by saying that it was self-evident that God had used mathematics in creating the world. Some sections of the Church, however, particularly the Protestants, objected to the description proposed by Copernicus. One of their objections was based on the biblical sentence (in Joshua), “Sun, stand still in Givon.” If the Sun was stationary in any case, they claimed, there was no need to command it to stand still. Copernicus did not accept that criticism passively but answered firmly in a letter to Pope Paul III that someone who is ignorant in mathematics cannot judge a mathematical theory, and the holy scriptures teach us how to get to heaven but not how the heavens are constructed. Despite the assertiveness of his letter, the opposition of significant parts of the Church stopped Copernicus from publishing his findings sooner. The book containing his complete theory was submitted to the printers in 1543, and the first copy reached Copernicus when he lay on his deathbed, a few days before he died on May 24, 1543.

Copernicus's model adopted Ptolemy's mathematical method but amended the mathematics to be consistent with the idea that the Sun was at the center of the universe and that the planets revolved around it. Copernicus accepted the principle that dated back to the Pythagoreans, that celestial motion had to be along perfect circles, and therefore the planets revolve along circles around the Sun or on circles that move around a center that itself revolves on a circle around the Sun, that is, on an epicycle. Copernicus made sophisticated use of Ptolemy's mathematical systems, and his greatest achievement, in his own words, was to reach a level of accuracy similar to Ptolemy's but by using only thirty-four epicycles and deferents. Yet, to achieve accurate results, Copernicus located the Sun only close to the center of the deferent, and not at its center, in the same way as Apollonius the Greek had located the Earth in his geocentric model. The desire for simplicity was one of Copernicus's reasons for his faith in his model. One of the arguments he used was that God would not have chosen to use seventy-seven orbits when thirty-four were sufficient.

However, the quest for simplicity and aestheticism also had the effect of halting progress. Copernicus was convinced that for reasons of perfection and aestheticism, the orbits of the stars had to be circular. God could
not and would not have created a world in which the paths of the stars were not perfect, that is to say, were not circles.

16. GIANTS’ SHOULDERS

It was Newton who developed modern mathematics to describe nature. His response to the praise he received was, “I stood on the shoulders of giants.” The circumstances in which he stated this do not prove the extent of his modesty or magnanimity, as we shall see further on, but the giants to whom he referred, Galileo Galilei, René Descartes, and Johannes Kepler, did make enormous contributions to mathematics and to understanding nature. In this section we will discuss the contribution of the first two to Newton's theory, and we will deal with Kepler in the next section.

Galileo Galilei was born in 1564 in Pisa, Italy, and died while under house arrest in his home in Florence in 1642. His family intended him to study religion and medicine, but he did not persevere in those fields and focused on the study of nature and mathematics, and he also exhibited a flair for commerce. He built a telescope following a Dutch invention that came to his attention and offered it to the city council of Venice for a generous annual stipend, to be used for early detection of enemy ships threatening the city. He showed the same flair and talent for science. He was the first to direct the telescope toward the heavens and the first to discover that the solar system consisted of more than the celestial bodies known to the Greeks. He discovered the four large moons of Jupiter, and his sharp political instincts led him to name them after members of the Medici family, the family of the Grand Duke of Tuscany, who later became Galileo's patron. He also used his telescope to study the surface of the Moon, realizing that the Moon had mountains and valleys, and he even calculated the height of the mountains. He also proved that the Moon was illuminated by light reflected from the Earth, and the directions of this reflected light deepened his faith in Copernicus's model. Galileo also tried to find additional proof supporting the heliocentric model. One of his “proofs” was an explanation
of the ebb and flow of the tide. He claimed that they occurred as a result of the Sun drawing the water toward itself at the same time as the Earth is revolving around its own axis. This explanation is incorrect, because if it were correct, high tide would occur just once a day and not twice as actually happens. Galileo was so eager to confirm the heliocentric model that he found an “excuse” for the discrepancy in his explanation. He also rejected Kepler's explanation that the cause of the tides was the Moon. The full explanation was not revealed until the days of Newton, and it will be discussed in a later section. By using his telescope, Galileo also discovered the phases of the planet Venus, which varied from a full circle to a thin crescent that disappeared for a short time, similar to the way we see the Moon. This strengthened Galileo's opinion of the heliocentric universe. His findings made him famous all over the European continent, and he was known as a firm believer in Copernicus's model.