Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

by Alexander, Amir

  Clavius believed that he knew what this secret was: mathematics. Theological and philosophical disputes could rage forever, he believed, because there was no universally accepted way to decide who was right and who was wrong. Even when one side possessed the absolute truth (as Clavius believed it did), and the other nothing but error, the adherents of error could still refuse to accept the truth. But mathematics was different: with mathematics, the truth forces itself upon its audience whether they like it or not. One could dispute the Catholic doctrine of the sacraments, but one could not deny the Pythagorean theorem; and no one could challenge the correctness of the new calendar, based as it was on detailed mathematical calculations. Here, Clavius believed, was a key to the ultimate triumph of the Church.

  THE CERTAINTY OF MATHEMATICS

  Clavius elaborated his views on mathematics in an essay that he attached to his edition of Euclid, which first came out in 1574, just as the commission on the calendar was getting down to work. Entitled simply “In disciplinas mathematicas prolegomena” (“Introductory Essay on the Mathematical Sciences”), it is in fact a passionate appeal for recognition of the power of the mathematical sciences and their superiority over other disciplines. If “the nobility and the excellence of a science is to be judged by the certainty of the demonstrations that it uses,” Clavius wrote, then “without a doubt the mathematical disciplines have the first place among all others”: “They demonstrate everything in which they see a dispute by the strongest reasons, and they confirm it in such a way that they engender true knowledge in the minds of the hearers, and completely remove any doubt.” Mathematics, in other words, imposes itself on the minds of its hearers and compels even the most recalcitrant among them to accept its truths.

  “The theorems of Euclid,” he continues, “and the rest of the mathematicians,”

  still today as for many years past, retain in the schools their true purity, their real certitude, and their strong and firm demonstrations … And thus so much do the mathematical disciplines desire, esteem, and foster the truth that they reject not only whatever is false, but even anything merely probable, and they admit nothing that does not lend support and corroboration to the most certain demonstrations.

  But the case is very different with the other so-called “sciences.” Here, Clavius argues, the intellect deals with a “multitude of opinions” and a “variety of views on the truth of the conclusions that are being assessed.” The result is that whereas mathematics leads to certainty that ends all debate, other fields leave the mind confused and uncertain. Indeed, Clavius continues, commenting on the inherent inconclusiveness of nonmathematical fields, “how far all this is from mathematics, I think no one admits.” “There can be no doubt,” he concludes, “but that the first place among the other sciences should be conceded to mathematics.”

  Rigorous, orderly, and irresistible, mathematics was for Clavius the embodiment of the Jesuit program. By imposing truth and vanquishing error, it established a fixed order and certainty in place of chaos and confusion. It should be remembered, however, that when Clavius is speaking of “mathematics,” he has something quite specific in mind. Certainly the arithmetic in use by merchants and traders had its place, as did the emerging new science of algebra, which teaches one how to solve quadratic, cubic, and quartic equations. But the true model of mathematical perfection for Clavius was geometry, as presented in Euclid’s great opus The Elements. It was the only mathematical field, he believed, that captured the power and truth of the discipline in its most distilled form. When Clavius wished to emphasize the eternal truth of mathematics, he cited “the demonstrations of Euclid,” and surely it is no coincidence that of all his textbooks on the many mathematical fields, he chose to append his “Prolegomena” to his edition of Euclid.

  Composed around 300 BCE, The Elements is arguably the most influential mathematical text in history. But not because it presented new and original results: The Elements was based on the work of earlier generations of geometers, and most of its results were likely well known to practicing mathematicians. What was revolutionary about Euclid’s work was its systematic and rigorous method. It begins with a series of definitions and postulates that are so simple as to be self-evidently true. According to one definition, “A figure is that which is contained by any boundary or boundaries”; according to one postulate, “all right angles are equal to one another”; and so on. From these seemingly trivial beginnings, Euclid moves step by step to demonstrate increasingly complex results: that the base angles of an isosceles triangle are equal; that in a right triangle, the sum of the squares of the two sides containing the right angle is equal to the square of the third side (the Pythagorean theorem); that in a circle, the angles in the same segment are all equal to one another; and so on. At each step, Euclid does not just argue that his result is plausible or likely, but demonstrates that it is absolutely true and cannot be otherwise. In this manner, layer by layer, Euclid constructs an edifice of mathematical truth, composed of interconnected and unshakably true propositions, each dependent on the ones that precede it. As Clavius points out in the “Prolegomena,” it was the sturdiest edifice in the kingdom of knowledge.

  For a taste of the Euclidean method, consider Euclid’s proof of proposition 32 in book 1: that the sum of the angles of any triangle is equal to two right angles—or, as we would say, 180 degrees. Euclid, at this point, has already proven that when a straight line falls on two parallel lines, it creates the same angles with one parallel line as with the other (book 1, proposition 29). He makes good use of this theorem here:

  Proposition 32: In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to two right angles.

  Proof:

  Let ABC be a triangle and let one side of it be produced to D. I say that the exterior angle ACD is equal to the two interior and opposite angles CAB, ABC, and the three interior angles of the triangle, ABC, BCA, CAB, are equal to two right angles.

  Figure 2.1. The sum of the angles in a triangle.

  For let CE be drawn through the point C, parallel to the line AB.

  Then, since AB is parallel to CE and AC falls on both of them, the alternate angles BAC and ACE are equal to one another.

  Again, since AB is parallel to CE, and the straight line BD falls on them, the exterior angle ECD is equal to the interior and opposite angle ABC.

  But the angle ACE was also proved equal to BAC. It follows that the whole angle ACD (composed of ACE and ECD) is equal to the two interior and opposite angles BAC and ABC.

  Let the angle ACB be added to each; it follows that the sum of the angles ACB and ACD are equal to the sum of the interior angles of the triangle, ABC, BCA, CAB.

  But since the angles ACB and ACE are equal to two right angles, it follows that the angles of the triangle, ABC, BCA, CAB are also equal to two right angles.

  Q.E.D.

  Euclid’s proof here is fundamentally simple: He extends the triangle’s side BC to the point D, and then draws a parallel line to AB through the opposite corner C. Using what he has already proven about the properties of parallels, he transfers the triangle’s angles A and B to the line BD next to the angle C, thus showing that the three angles together combine to form a straight line—that is, 180 degrees. But even in this simple proof, all the elements that make Euclid so compelling are clearly present. The proof is based on previous ones, in this case, the unique properties of parallels; from there, it proceeds systematically, step by step, showing clearly that each small step is logically correct and necessary; and ultimately, it arrives at its conclusion, which is absolutely true and universal. Not only the specific triangle ABC has angles that combine to 180 degrees, but every triangle that ever was, will be, or can be will show the exact same characteristic. Finally, the proof of proposition 32 and every other Euclidean proof is a microcosm of Euclid’s geometry as a whole. Just as each proof is composed of small logical
steps, so the proofs themselves are but small steps in the edifice that is Euclidean geometry. And like each proof alone, geometry as a whole is universally and eternally true, ordering the world and governing its structure everywhere and always.

  It was clear to Clavius that Euclid’s method had succeeded in doing precisely what the Jesuits were struggling so hard to accomplish: imposing a true, eternal, and unchallengeable order upon a seemingly chaotic reality. The diverse world we see around us, made of seemingly limitless shapes, colors, and textures, might appear to us as chaotic and unruly. But thanks to Euclid, we know better: all this diversity and apparent chaos is in fact strictly ordered by the eternal and universal truths of geometry. Antonio Possevino, a Jesuit, papal nuncio (ambassador), and friend and collaborator of Clavius, made this point in his Bibliotheca selecta of 1591, where he argues that

  if anyone mentally conceives of God as wisest and as Geometrical architect for all … he will understand that the world had been joined by God from all substances and from the whole of matter; but since he wished to leave nothing discordant and unordered, but to adorn it with ratio, measurement, and number … therefore the Craftsman of the world imitated the fairest and eternal exemplar.

  God had imposed geometry upon unruly matter, and hence the eternal rules of geometry prevail everywhere and always.

  Mathematics, and geometry in particular, was for Clavius an expression of the highest Jesuit ideals and provided a clear road map for the Society as it struggled to build a new Catholic order. In some instances mathematics could be used directly to enhance the power of the Church, as was the case with the reform of the calendar. In other instances mathematics could serve as an ideal model for true knowledge, which the other disciplines could aspire to emulate. Either way, to Clavius one thing was clear: mathematics could no longer languish as an afterthought in the Jesuit empire of learning, but must become a core discipline of the curriculum and a key component in the formation of Jesuits.

  CLAVIUS AGAINST THE THEOLOGIANS

  The road to establishing mathematics as a core discipline in the Jesuit curriculum was a difficult one. In the first place, Clavius had to deal with those of his colleagues who simply did not believe that mathematics deserved the high position in which he wished to place it. Ignatius, they pointed out, had not placed much stock in mathematics, and the authorities he had prescribed were not particularly favorable to mathematics. Aquinas, Ignatius’s chosen theological authority, had only limited use for simple mathematics; Aristotle, the Jesuits’ guide in philosophy, assigned mathematics a far smaller role than did his teacher and philosophical rival Plato; and in Aristotelian physics and biology, mathematics played no part at all.

  The most outspoken of Clavius’s opponents at the Collegio Romano seems to have been the theologian Benito Pereira, the same Jesuit who had proclaimed that one must always “adhere to the old and generally accepted opinions.” “My opinion,” Pereira declared in 1576, just as Clavius was launching into the project of calendar reform, “is that mathematical disciplines are not proper sciences.” The problem with mathematics, according to Pereira, is that its demonstrations are weak, and consequently, it does not produce true knowledge, referred to in the philosophical language of the time as scientia. This is because proper demonstrations, according to Aristotle, proceed from true causes—those rooted in the essential nature of the objects discussed. For example, the classic syllogism

  All men are mortal

  Socrates is a man

  Therefore Socrates is mortal

  proceeds from the fact that mortality is an essential part of being human. But nothing like this, Pereira argues, exists in mathematics, because mathematical demonstrations do not take into account the essence of things. Instead, they point to complex relations between numbers, lines, figures, etc.—all interesting in themselves, no doubt, but lacking the logical force of a demonstration from true causes. The use of parallel lines, for example, might reveal to us that the sum of the angles of a triangle is equal to two right angles, but the parallel lines did not cause this to be true. For all intents and purposes, Pereira suggests, mathematics doesn’t even have a true subject matter; it merely draws connections between different properties. If one seeks strong demonstrations, one must turn elsewhere: to the syllogistic demonstrations of Aristotelian physics, which are almost entirely devoid of mathematics.

  Not so, retorted Clavius in the “Prolegomena.” The subject of mathematics is matter itself, since all mathematics is “immersed” in matter. This, he argues, puts mathematics in a distinguished place in the order of knowledge: both immersed in matter and abstracted from it, mathematics is halfway between physics, which deals only with matter, and metaphysics, which deals with things separated from matter. Mathematics, according to Clavius, should not aspire to equality with metaphysical theology, which deals with things such as the soul and salvation. But it is, nonetheless, clearly in a superior position to the Aristotelian physics favored by Pereira. Whether Clavius won the argument is a matter of opinion. Contemporaries thought that he at least held his own, and that is really all he needed. His rising prestige as the Society’s representative on the calendar commission did more than his logical and rhetorical powers to bolster his arguments, and in any case, he was more interested in actual pedagogical reform than in abstract philosophical debate. That is where he directed his fight, and that is where he would ultimately win it.

  Clavius laid out his plans for raising the profile of mathematics in the Society in a document called “Modus quo disciplinas mathematicas in scholis Societatis possent promoveri” (“The ways in which the mathematical disciplines could be promoted in the Society’s schools”), which he circulated around 1582, shortly after the calendar commission had completed its work. In order for the program to succeed, he argued, it was first necessary to raise the prestige of the field in the eyes of the students. This would require some cooperation from his colleagues, and he did not hesitate to take direct aim at those he suspected of sabotaging his efforts. He clearly had Pereira and his allies in mind when he complained that reliable sources had informed him that certain teachers openly mocked the mathematical sciences. “It will contribute much,” he wrote, to the promotion of mathematics

  if the teachers of philosophy abstain from those questions which do not help in the understanding of natural things and very much detract from the authority of the mathematical disciplines in the eyes of students, such as those in which they teach that the mathematical sciences are not sciences [and] do not have demonstrations …

  “Experience teaches,” he added acidly, “that these questions are a great hindrance to students and of no service to them.”

  Apart from countering the pernicious influence of hostile colleagues, Clavius also made positive suggestions for the advancement of mathematics in the Society’s schools. First and foremost, he argued, master teachers must be found “with uncommon erudition and authority,” since without those, students “seem unable to be attracted to the mathematical disciplines.” In order to produce a cadre of such capable professors, Clavius suggested establishing a special school, where the most promising mathematics students in the Jesuit colleges would be sent to pursue higher studies. Later on, once they took up their regular teaching positions, the graduates of the school “should not be taken up with many other occupations,” but be left to focus on mathematical instruction. To counter antimathematical prejudice, it was extremely important that these highly trained mathematicians be treated by their colleagues with the utmost respect, and invited to take part in public disputations alongside the professors of theology and philosophy. The prestige of mathematics, he explained, required this: “pupils up to now seem almost to have despised these sciences for the simple reason that they think that they are not considered of value and are even useless, since the person who teaches them is never summoned to public acts with the other professors.”

  Then as now, students were very quick to pick up on which subjects and teachers
were valued and which were not, and it was close to impossible for instructors in an undervalued field to get the students to take them seriously. Today it is more likely to be teachers of philosophy and the humanities who complain that their fields are disrespected by instructors in the prestigious mathematical sciences. But even if the roles of the different disciplines are roughly reversed today, the dynamic is still much the same.

  THE EUCLIDEAN KEY

  Staffing the colleges with qualified teachers was one thing. Giving them something to teach, however, was another, and here again Clavius stepped in with a proposal. Already in 1581 he wrote up a detailed mathematical curriculum, which he called “Ordo servandus in addiscendis disciplinis mathematicis”—literally, “The order to be kept in learning the mathematical disciplines.” His complete curriculum consisted of twenty-two lesson sets spread over three years of study, a plan that ultimately proved too ambitious to be generally implemented. In the Jesuit colleges, theology and philosophy still came first. Nevertheless, this did not prevent Clavius from pushing hard to introduce as much of his curriculum as possible into the schools.

  The first, most important, and key component of Clavius’s curriculum was inevitably Euclidean geometry. Any incoming student would start out by studying the first four books of Euclid, which deal with plane geometry. He would then study the fundamentals of arithmetic, before moving on to astronomy, geography, perspective, and music theory, among others, each according to the accepted authority on the subject: Jordanus de Nemore on arithmetic, Sacrobosco on astronomy, Ptolemy on geography, and so on. But he would return time and again to the greatest master of the mathematical sciences, Euclid, until he had thoroughly mastered the entire thirteen books of The Elements. It was a logical sequence of studies, but for Clavius it also represented a deeper ideological commitment. Geometry, being rigorous and hierarchical, was, to the Jesuit, the ideal science. The mathematical sciences that followed—astronomy, geography, perspective, music—were all derived from the truths of geometry, and demonstrated how those truths governed the world. Consequently, Clavius’s mathematical curriculum did not just teach the students specific competencies. More important, it demonstrated how absolute eternal truths shape the world and govern it.