Is God a Mathematician? (11 page)
Read Is God a Mathematician? Online
Authors: Mario Livio
Figure 19
Galileo never gave up an opportunity for a good fight. His most articulate exposition of his thoughts on the nature of mathematics and of its role in science appears in another polemic publication—
The Assayer
. This brilliant, masterfully written treatise became so popular that Pope Urban VIII had pages from it read to him during his meals. Oddly enough, Galileo’s central thesis in
The Assayer
was patently wrong. He tried to argue that comets were really phenomena caused by some quirks of optical refraction on this side of the Moon.
The entire story of
The Assayer
sounds a bit as if it were taken from the libretto of an Italian opera. In the fall of 1618, three comets appeared in succession. The third one, in particular, remained visible for almost three months. In 1619, Horatio Grassi, a mathematician from the Jesuit Collegio Romano, anonymously published a pamphlet about his observations of these comets. Following in the footsteps of the great Danish astronomer Tycho Brahe, Grassi concluded that the comets were somewhere between the Moon and the Sun. The pamphlet might have gone unnoticed, but Galileo decided to respond, having been told that some Jesuits took Grassi’s publication as a blow to Copernicanism. His reply was in the form of lectures (largely written by Galileo himself) that were delivered by Galileo’s disciple Mario Guiducci. In the published version of these lectures,
Discourse on the Comets,
Galileo directly attacked Grassi and Tycho Brahe. This time it was Grassi’s turn to take offense. Under the pseudonym of Lothario Sarsi, and posing as one of his own students, Grassi published an acrimonious reply, criticizing Galileo in no uncertain terms (the response was entitled
The Astronomical and Philosophical Balance, on which the opinions of Galileo Galilei regarding Comets are weighed, as well as those presented in the Florentine Academy by Mario Guiduccio
). In defense of his application of Tycho’s methods for determining distances, Grassi (speaking as if he were his student) argued:
Let it be granted that my master followed Tycho. Is this such a crime? Whom instead should he follow? Ptolemy [the Alexandrian originator of the heliocentric system]? Whose follow
ers’ throats are threatened by the out-thrust sword of Mars now made closer. Copernicus? But he who is pious will rather call everyone away from him and will spurn and reject his recently condemned hypothesis. Therefore, Tycho remains as the only one whom we may approve of as our leader among the unknown courses of the stars.
This text beautifully demonstrates the fine line that Jesuit mathematicians had to walk at the beginning of the seventeenth century. On one hand, Grassi’s criticism of Galileo was entirely justified and penetratingly insightful. On the other, by being forced not to commit to Copernicanism, Grassi had imposed upon himself a straitjacket that impaired his overall reasoning.
Galileo’s friends were so concerned that Grassi’s attack would undermine Galileo’s authority that they urged the master to respond. This led to the publication of
The Assayer
in 1623 (the full title explained that in the document “are weighed with a fine and accurate balance the contents of the
Astronomical and Philosophical Weighing Scales
of Lothario Sarsi of Siguenza”).
As I noted above,
The Assayer
contains Galileo’s clearest and most powerful statement concerning the relation between mathematics and the cosmos. Here is that remarkable text:
I believe Sarsi is firmly convinced that it is essential in philosophy to support oneself by the opinion of some famous author, as if when our minds are not wedded to the reasoning of someone else they ought to remain completely barren and sterile. Perhaps he thinks that philosophy is a book of fiction created by some man, like the
Iliad
or
Orlando Furioso
[an epic sixteenth century poem by Ludovico Ariosto]—books in which the least important thing is whether what is written in them is true. Sig. Sarsi, this is not how matters stand.
Philosophy is written in that great book which ever lies before our eyes (I mean the universe) but we cannot understand it if we do not first learn the language and grasp the characters in which it is written. It is written in the language of mathematics, and the
characters are triangles, circles and other geometrical figures, without which it is humanly impossible to comprehend a single word of it, and without which one wanders in vain through a dark labyrinth.
[emphasis added]
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 Congregation 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.