Is God a Mathematician?

by Mario Livio

  Mind-boggling, isn’t it? Centuries before the question of why mathematics was so effective in explaining nature was even asked, Galileo thought he already knew the answer! To him, mathematics was simply the language of the universe. To understand the universe, he argued, one must speak this language. God is indeed a mathematician.

  The full range of ideas in Galileo’s writings paints an even more detailed picture of his views on mathematics. First, we must realize that to Galileo, mathematics ultimately meant geometry. Rarely was he interested in measuring values in absolute numbers. He described phenomena mainly with proportions among quantities and in relative terms. In this again, Galileo was a true disciple of Archimedes, whose principle of the lever and method of comparative geometry he used effectively and extensively. A second interesting point, which is revealed especially in Galileo’s last book, is the distinction he makes between the roles of geometry and logic. The book itself, Discourses and Mathematical Demonstrations Concerning Two New Sciences, is written in the form of lively discussions among three interlocutors, Salviati, Sagredo, and Simplicio, whose roles are quite clearly demarcated. Salviati is effectively Galileo’s spokesman. Sagredo, the aristocratic philosophy lover, is a man whose mind has already escaped from the illusions of Aristotelian common sense and who can therefore be persuaded by the strength of the new mathematical science. Simplicio, who in Galileo’s previous work was portrayed as being under the spell of Aristotelian authority, appears here as an open-minded scholar. On the second day of the argument, Sagredo has an interesting exchange with Simplicio:

  Sagredo: What shall we say, Simplicio? Must we not confess that the power of geometry is the most potent instrument of all to sharpen the mind and dispose it to reason perfectly, and to speculate? Didn’t Plato have good reason to want his pupils to be first grounded in mathematics?

  Simplicio appears to agree and he introduces the comparison with logic:

  Simplicio: Truly I begin to understand that although logic is a very excellent instrument to govern our reasoning, it does not compare with the sharpness of geometry in awakening the mind to discovery.

  Sagredo then sharpens the distinction:

  Sagredo: It seems to me that logic teaches how to know whether or not reasoning and demonstrations already discovered are conclusive, but I do not believe that it teaches how to find conclusive reasoning and demonstrations.

  Galileo’s message here is simple—he believed that geometry was the tool by which new truths are discovered. Logic, on the other hand, was to him the means by which discoveries already made are evaluated and critiqued. In chapter 7 we shall examine a different perspective, according to which the whole of mathematics stems from logic.

  How did Galileo arrive at the idea that mathematics was nature’s language? After all, a philosophical conclusion of this magnitude could not have suddenly materialized out of thin air. Indeed, the roots of this conception can be traced all the way back to the writings of Archimedes. The Greek master was the first to use mathematics to explain natural phenomena. Then, via a tortuous path passing through some medieval calculators and Italian court mathematicians, the nature of mathematics gained the status of a subject worthy of discussion. Eventually, some of the Jesuit mathematicians of Galileo’s time, Christopher Clavius in particular, also acknowledged the fact that mathematics might occupy some middle ground between metaphysics—the philosophical principles of the nature of being—and physical reality. In the preface (“Prolegomena”) to his Comments on Euclid’s “Elements,” Clavius wrote:

  Since the mathematical disciplines deal with things which are considered apart from any sensible matter, although they are immersed in material things, it is clear that they hold a place intermediate between metaphysics and natural science, if we consider their subject matter.

  Galileo was not satisfied with mathematics as the mere go-between or conduit. He took the extra bold step of equating mathematics with God’s native tongue. This identification, however, raised another serious problem—one that was about to have a dramatic impact on Galileo’s life.

  Science and Theology

  According to Galileo, God spoke in the language of mathematics in designing nature. According to the Catholic Church, God was the “author” of the Bible. What was one to make then of those cases where the mathematically based scientific explanations seemed to contradict the scriptures? The theologians of the 1546 Council of Trent answered in no uncertain terms: “No one relying on his own judgment and distorting the Sacred Scriptures according to his own conception shall dare to interpret them contrary to that sense which Holy Mother Church, to whom it belongs to judge of their true sense and meaning, has held or does hold.” Accordingly, when in 1616 theologians were asked to give their opinion on Copernicus’s heliocentric cosmology, they concluded that it was “formally heretical, since it explicitly contradicts in many places the sense of the Holy Scripture.” In other words, what was truly at the heart of the Church’s objection to Galileo’s Copernicanism was not so much the removal of the Earth from its central position in the cosmos, but rather the challenge to the church’s authority in interpreting the scriptures. In a climate in which the Roman Catholic Church was already feeling embattled by controversies with Reformation theologians, Galileo and the Church were on a clear collision course.

  Events started to unfold rapidly toward the end of 1613. Galileo’s former student, Benedetto Castelli, made a presentation of the new astronomical discoveries to the grand duke of Tuscany and his entourage. Predictably, he was pressured to explain the apparent discrepancies between the Copernican cosmology and some biblical accounts, such as the one in which God stopped the Sun and the Moon in their courses to allow Joshua and the Israelites to complete their victory over the Emorites in the Ayalon Valley. Even though Castelli reported that he “behaved like a champion” in defending Copernicanism, Galileo was somewhat disturbed by the news of this confrontation, and he felt compelled to express his own views about contradictions between science and the Holy Scriptures. In a long letter to Castelli dated December 21, 1613, Galileo writes:

  It was necessary, however in the Holy Scripture, in order to accommodate itself to the understanding of the majority, to say many things which apparently differ from the precise meaning. Nature, on the contrary, is inexorable and unchangeable, and cares not whether her hidden causes and modes of working are intelligible to the human understanding or not, and never deviates on that account from the prescribed laws. It appears to me therefore that no effect of nature, which experience places before our eyes, or is the necessary conclusion derived from evidence, should be rendered doubtful by passages of Scripture which contain thousands of words admitting of various interpretations, for every sentence of Scripture is not bound by such rigid laws as is every effect of nature.

  This interpretation of the biblical meaning was clearly at odds with that of some of the more stringent theologians. For instance, the Dominican Domingo Bañez wrote in 1584: “The Holy Spirit not only inspired all that is contained in the Scripture, he also dictated and suggested every word with which it was written.” Galileo was obviously not convinced. In his Letter to Castelli he added:

  I am inclined to think that the authority of Holy Scripture is intended to convince men of those truths which are necessary for their salvation, and which being far above man’s understanding cannot be made credible by any learning, or any other means than revelation by the Holy Spirit. But that the same God that has endowed us with senses, reason, and understanding, does not permit us to use them, and desires to acquaint us in any other way with such knowledge as we are in a position to acquire for ourselves by means of those faculties, that it seems to me I am not bound to believe, especially concerning those sciences about which the Holy Scripture contains only small fragments and varying conclusions; and this is precisely the case with astronomy, of which there is so little that the planets are not even all enumerated.

  A copy of Galileo’s letter made it to the Congregatio
n of the Holy Office in Rome, where affairs concerning faith were commonly evaluated, and especially to the influential Cardinal Robert Bellarmine (1542–1621). Bellarmine’s original reaction to Copernicanism was rather moderate, since he regarded the entire heliocentric model as “a way to save the appearances, in the manner of those who have proposed epicycles but do not really believe in their existence.” Like others before him, Bellarmine too treated the mathematical models put forth by astronomers as merely convenient gimmicks, designed to describe what humans observed, without being anchored in any physical reality. Such “saving the appearances” devices, he argued, do not demonstrate that the Earth is really moving. Consequently, Bellarmine saw no immediate threat from Copernicus’s book (De Revolutionibus), even though he was quick to add that to claim that the Earth was moving would not only “irritate all scholastic philosophers and theologians” but would also “harm the Holy Faith by rendering Holy Scripture false.”

  The full details of the rest of this tragic story are beyond the scope and main focus of the present book, so I’ll describe them only briefly here. The Congregation of the Index banned Copernicus’s book in 1616. Galileo’s further attempts to rely on numerous passages from the most revered of the early theologians—St. Augustine—to support his interpretation of the relation between the natural sciences and Scripture did not gain him much sympathy. In spite of articulate letters in which his main thesis was that there is no disagreement (other than superficial) between the Copernican theory and the biblical texts, the theologians of his day regarded Galileo’s arguments as an uninvited foray into their domain. Cynically, these same theologians did not hesitate to express opinions on scientific matters.

  As the dark clouds were gathering on the horizon, Galileo continued to believe that reason would prevail—a huge mistake when it comes to challenging faith. Galileo published his Dialogue Concerning the Two Chief World Systems in February of 1632 (figure 20 shows the frontispiece of the first edition). This polemical text was Galileo’s most detailed exposition of his Copernican ideas. Moreover, Galileo argued that by pursuing science using the language of mechanical equilibrium and mathematics, humans could understand the divine mind. Put differently, when a person finds a solution to a problem using proportional geometry, the insights and understanding gained are godlike. The church’s reaction was swift and decisive. The circulation of the Dialogue was forbidden as early as August of the year of its publication. In the following month, Galileo was summoned to Rome to defend himself against the charges of heresy. Galileo was brought to trial on April 12, 1633, and he was found “vehemently suspect of heresy” on June 22, 1633. The judges accused Galileo “of having believed and held the doctrine—which is false and contrary to the sacred and divine Scriptures—that the Sun is the center of the world and does not move from east to west and that the Earth moves and is not the center of the world.” The sentence was harsh:

  We condemn you to the formal prison of this Holy Office during our pleasure, and by way of salutary penance we enjoin that for three years to come you repeat once a week the seven penitential Psalms. Reserving to ourselves liberty to moderate, commute, or take off, in whole or in part, the aforementioned penalties and penance.

  The devastated seventy-year-old Galileo could not withstand the pressure. His spirit broken, Galileo submitted his letter of abjuration, in which he committed to “abandon completely the false opinion that the Sun is at the center of the world and does not move and that the Earth is not the center of the world and moves.” He concluded:

  Figure 20

  Therefore, desiring to remove from the minds of your Eminences, and of all faithful Christians, this vehement suspicion justly conceived against me, with sincere heart and unfeigned faith I abjure, curse, and detest the aforesaid errors and heresies and generally every other error, heresy, and sect whatsoever contrary to the Holy Church, and I swear that in future I will never again say or assert, verbally or in writing, anything that might furnish occasion for a similar suspicion regarding me.

  Galileo’s last book, Discourses and Mathematical Demonstrations Concerning Two New Sciences, was published in July 1638. The manuscript was smuggled out of Italy and was published in Leiden in Holland. The content of this book truly and powerfully expressed the sentiment embodied in the legendary words “Eppur si muove” (“And yet it moves”). That defiant phrase, commonly put in Galileo’s mouth at the end of his trial, was probably never uttered.

  On October 31, 1992, the Catholic Church finally decided to “rehabilitate” Galileo. Recognizing that Galileo was right all along, but still avoiding direct criticism of the Inquisition, Pope John Paul II said:

  Paradoxically, Galileo, a sincere believer, proved himself more perspicacious on this issue [apparent discrepancies between science and the scriptures] than his theologian adversaries. The majority of theologians did not perceive the formal distinction that exists between the Holy Scripture in itself and its interpretation, and this led them unduly transferring to the field of religious doctrine an issue which actually belongs to scientific research.

  Newspapers around the world had a feast. The Los Angeles Times declared: “It’s Official: The Earth Revolves Around the Sun, Even for the Vatican.”

  Many were not amused. Some saw this mea culpa by the church as far too little, far too late. The Spanish Galileo scholar Antonio Beltrán Marí noted:

  The fact that the Pope continues to consider himself an authority capable of saying something relevant about Galileo and his science shows that, on the Pope’s side, nothing has changed. He is behaving in exactly the same manner as Galileo’s judges, whose mistake he now recognizes.

  To be fair, the Pope found himself in a no-win situation. Any decision on his part, whether to ignore the issue and keep Galileo’s condemnation on the books, or to finally acknowledge the church’s error, was likely to be criticized. Still, at a time when there are attempts to introduce biblical creationism as an alternative “scientific” theory (under the thinly veiled title of “intelligent design”), it is good to remember that Galileo already fought this battle almost four hundred years ago—and won!

  CHAPTER 4

  MAGICIANS: THE SKEPTIC AND THE GIANT

  In one of the seven skits in the movie Everything You Always Wanted to Know About Sex* (*But Were Afraid to Ask), Woody Allen plays a court jester who does comic routines for a medieval king and his entourage. The jester has the hots for the queen, so he gives her an aphrodisiac, hoping to seduce her. The queen does become attracted to the jester, but alas, she has a huge padlock on her chastity belt. Faced with this frustrating situation in the queen’s bedroom, the jester utters nervously: “I must think of something quickly, before the Renaissance will be here and we will all be painting.”

  Jokes aside, this exaggeration is an understandable description of the events in Europe during the fifteenth and sixteenth centuries. The Renaissance has indeed produced such a wealth of masterpieces in painting, sculpture, and architecture that to this very day, these astonishing works of art form a major part of our culture. In science, the Renaissance witnessed the heliocentric revolution in astronomy, led by Copernicus, Kepler, and especially Galileo. The new view of the universe afforded by Galileo’s observations with the telescope, and the insights gained from his experiments in mechanics, perhaps more than anything else motivated the mathematical developments of the following century. Amidst the first signs of crumbling of the Aristotelian philosophy and the challenges to the Church’s theological ideology, philosophers started to search for a new foundation on which to erect human knowledge. Mathematics, with its seemingly certain body of truths, provided what appeared to be the soundest base for a new start.

  The man who embarked on the rather ambitious task of discovering a formula that would somehow discipline all rational thought and unify all knowledge, science, and ethics was a young French officer and gentleman named René Descartes.

  A Dreamer

  Many regard Descartes (figure 21) a
s both the first great modern philosopher and the first modern biologist. When you add to these impressive credentials the fact that the English empiricist philosopher John Stuart Mill (1806–73) characterized one of Descartes’ achievements in mathematics as “the greatest single step ever made in the progress of the exact sciences,” you begin to realize the immensity of Descartes’ power of intellect.

  René Descartes was born on March 31, 1596, at La Haye, France. In honor of its most celebrated resident, the town was renamed La Haye–Descartes in 1801, and since 1967 it is known simply as Descartes. At the age of eight, Descartes entered the Jesuit College of La Flèche, where he studied Latin, mathematics, classics, science, and scholastic philosophy until 1612. Because of his relatively fragile health, Descartes was excused from having to get up at the brutal hour of five a.m., and he was allowed to spend the morning hours in bed. Later in life, he continued to use the early part of the day for contemplation, and he once told the French mathematician Blaise Pascal that the only way for him to stay healthy and be productive was to never get up before he felt comfortable doing so. As we shall soon see, this statement turned out to be tragically prophetic.