Many moons ago when I studied mathematics at university in Germany, the first two years (four semesters) essentially consisted of two ninety minute lectures, starting at eight o’clock, four days a week covering the basics in a continuous stream of theorem, proof, theorem, proof, theorem, lemma, proof, corollary, theorem, proof for ninety straight minutes. Each lecture course had an extensive exercise sheet of problems to be solved, with a seminar on those problems in the afternoon. It was a hard slog. Of the roughly two hundred and fifty student in the lecture theatre more than half were studying physics and not mathematics. Of the remaining mathematics students, the majority were training to be gymnasium, read highschool teachers, diploma, read masters, mathematicians, like myself, making up a small minority. A couple of years later the physics department set up their own introductory courses for their students, arguing that the mathematics departments courses were too abstract and the physicists need a more applied approach.
If you look at modern physics most of it is extremely mathematical. Classical mechanics is all analysis, generally relativity theory is tensor calculus, quantum mechanics is vector spaces and so on and so forth. Just to annoy physicists, it would not be too much of an exaggeration to claim that modern physics is in reality all just applied mathematics. A view strongly supported by those who take an instrumentalist view of the theories of physics. If the maths works, who cares about the interpretation.
A central element of the classical version of the so-called scientific revolution was the mathematisation of science during this period. What this actually meant was the mathematisation of physics, a view that completely ignored the significance of major developments in a wide range of disciplines during the early modern period, history for example. However, in terms of this series, which sets out to trace the evolution of the discipline of physics from Aristotle’s ta physika down to modern physics that mathematisation is a very significant factor. One is tempted to call it a turning point but as it was spread out over a long period of time that would be a misleading term for what took place.
What happened here, why was the mathematisation even necessary? For Aristotle ta physika is an empirical description of nature and its laws. However, for Aristotle the objects of mathematics are purely abstract and thus do not exist in nature and so one can’t use them to describe nature. The move to using mathematics to describe nature is a major change in the presentation of knowledge. How and/or why did it take place? There is two widespread popular answers to this question, which are, however, in the one case far too simplistic and in the second simply wrong.
If you look at older texts on the history of science they explain that there was a fundamental shift in the philosophy underlying the study of nature in the sixteenth and seventeenth centuries. They state that the dominant Aristotelian philosophy was replaced by a Neo-Platonic philosophy. The Aristotelian philosophy rejected mathematics as a descriptive medium for nature, the Neo-Platonic philosophy with its roots in the Pythagorean mathematical world view was mathematics friendly. There was a major Neoplatonist revival during the Renaissance and several central Renaissance figures, such as Basilios Bessarion (1403–1472) and Marsilio Ficino (1433–1499), were dedicated to spreading a Platonic philosophy but this Neoplatonic movement was not particularly mathematical. Yes, there was a major Neoplatonic revival during the Renaissance but no it didn’t really lead to a mathematical approach to describing nature.
The other widespread story is the famous Galileo two books quote from Il Saggiatore (1623) concerning the Book of Scripture and the Book of Nature both written by God. On the Book of Nature, he wrote:
Philosophy is written in this grand book — I mean the universe — which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth.
An incredibly large number of people believe that Galileo was the first to utter this sentiment. That before him nobody was aware that nature can and should be understood using mathematics. This belief displays a horrible mixture of ignorance and hagiography. Before Galileo all was dark and he led us to the light! Or something like that.
Already in the thirteenth century the polymath Bishop of Lincoln, Robert Grosseteste (c. 1175–1253) wrote in his De lineis, angulis et figuris (On lines, angles and figures) between 1220 and 1235:
“…a consideration of lines, angles and figures is of the greatest utility because it is impossible to gain a knowledge of natural philosophy without them…for all causes of natural effects must be expressed by means of lines, angles and figures”
Moving much closer to home. When Galileo was an unknown wannabe student of mathematics he wrote a paper La Billancetta (The Little Balance) (1558) in which he presented a hydrostatic balance, which he hypothesised was the true method used by Archimedes to determine if that crown was really made of pure gold. He sent copies of this manuscript to two leading mathematicians, the aristocrat Guidobaldo dal Monte (1545–1607) and the Jesuit, Christoph Clavius (1538–1612). The professor of mathematics at the Collegio Romano, hoping to win one or both of them as a patron. He was successful with dal Monte, who used his connections to get him appointed professor of mathematics in Pisa, but it is Clavius who interests us here.
By the very nature of his Jesuit calling, Clavius was an Aristotelian Thomist, however from the very beginning of his appointment in Rome, he fought to get mathematics established as a prominent part of the Jesuit education programme. He met with a lot of opposition but was persistent and in the end won the battle. Clavius was convinced that the world could be described with mathematics and set up a training programme for mathematics teachers in the Collegio Romano, which turned out many of the best mathematicians in Europe in the seventeenth century. It was Clavius’ championing of mathematics, long before Galileo wrote his pathetic passage in Il Saggiatore, that led to Galileo sending him a copy of La Billancetta.
In what follows we will see that various others were applying mathematics to nature well before Galileo, but what led to them doing so? There were a series of contributing factors in the development.
Aristotle allowed the use of mathematics in the so called mixed sciences, astronomy, optics, and statics. In these areas the true nature of the phenomena under examination were explained philosophically so, the heavens by cosmology and vision by Aristotle’s theory of vision, theories that contain no mathematics. However, to be able to predict the movement of the planets or the laws of reflection one turns to mathematical astronomy or geometrical optics. According to Aristotle, these theories are purely practical and have no real interpretations. That is, they do not describe the real nature of the phenomena under examination.
This all began to change in the sixteenth century. Beginning with Copernicus (1473–1543) and continuing with Tycho Brahe (1546–1601) and Johannes Kepler (1571–1630) the astronomers rejected this dichotomy between descriptive cosmology and mathematical astronomy and stated quite clearly that their mathematical models were not merely calculating devices but represented reality. This was by no means accepted by all straight away but by the middle of the seventeenth century it was generally accepted that in astronomy the mathematical models represented reality.
The same process took place in optics. The sixteenth century saw advances in the study of optics that would lead to the theories of lenses, optics and vision of Francesco Maurolico (1494–1574), Giovanni Battista Della Porta (1535(?)–1615), Friedrich Risner (c. 1533–1580), reaching a high point in the optics of Johannes Kepler (1571–1630). In the optics of Kepler, his theories of vision and geometrical optics were integral parts of his total concept and the mathematics was now describing reality. As with astronomy the initial acceptance was slow but once again by the middle of the seventeenth century it was generally accepted that geometrical optics was describing reality.
A second important factor in the gradual introduction of mathematics into physics lay in the very nature of the Renaissance. What started out as a revival of the humanities from antiquity with emphasis on the language and the orators, Cicero for example, developed into a search for anything and everything over a wide range of topics. What little mathematics that was taught in the medieval universities, a theme we will return to, was basically Euclidian geometry. The Renaissance saw the revival of a wide range of mathematics including the engineering of Hero of Alexandria and Vitruvius, as well as the mathematics of Euclid, better quality sources and the first printed edition, and for the first time Apollonius and Archimedes. The works of Archimedes appeared in print for the first time in 1544. We have already seen how the renewed interest in the work of Archimedes played a significant role in the new developments in the laws of fall and hydrostatics. Mathematics was very much back on the menu.
As well as the mathematics from antiquity there was a strong influx of mathematics from the Islamicate cultures. Trigonometry, largely neglected in medieval Europe, saw a major revival through Arabic texts, reaching a high point with the publication of Regiomontanus’ De triangulis omnimodis a comprehensive introduction to both spherical and plane trigonometry in 1533 and his Tabulae Directionum in 1584 with its tangent tables missing from the earlier work.
Perhaps the most important Arabic contribution was algebra, the beginning of analytical mathematics. Already present in Europe since the thirteenth century through the Libre Abbaci of Leonardo Pisano (c. 1170–c. 1245), it had only really found a home outside the academic sphere as commercial arithmetic. This began to change with the German Cossists such as Christoff Rudolff (c.1500–before 1543), with his Behend und hübsch Rechnung durch die kunstreichen regeln Algebre, so gemeinicklich die Coß genennt werden (Deft and nifty reckoning with the artful rules of Algebra, commonly called the Coss), and his imitators such as Robert Recorde with his The whetstone of witte, whiche is the seconde parte of Arithmetike: containyng the extraction of Rootes: The Coßike practise, with the rule of Equation: and the woorkes of Surde Nombers, in 1557 and L’arithmétique by the Netherlander Simon Stevin (1548–1620) published in 1585. This raised the status of algebra but it still remained outside the academic sphere.
Algebra found access to the academic sphere through the Artis Magnae, Sive de Regulis Algebraicis Liber Unus (Book number one about The Great Art, or The Rules of Algebra) of Gerolamo Cardano (1501–1576) from 1545, the L’Algebra of Rafael Bombelli (1526–1572) from 1572, and the In artem analyticem isagoge of François Viète (1540–1603) from 1591.
I find it interesting for those who claim the Galileo was the great innovator in the use of mathematics in physics that he rejected the new analytical mathematics, sticking to his Euclidian geometry, whereas the Thomist, Aristotelian, Jesuit Cristoph Clavius wrote and published a textbook of Viète’s algebra for the use of his school and university mathematics teachers.
It is to the universities and their teaching of mathematics that we now turn. Although the undergraduate curriculum on the medieval European universities was nominally based on the trivium–grammar, rhetoric, dialectic–and the quadrivium–arithmetic, geometry, music, astronomy–in reality it was somewhat different. The trivium was taught largely on the basis of the analysis of the philosophical texts of Aristotle and his commentators. The quadrivium was, however, to a large extent neglected and in extreme cases simply ignored. Arithmetic was taught with the extremely elementary text of Boethius, geometry, of course, according to Euclid but whilst nominally restricted to the first six books, courses often got no further that Book I. Music was again Boethius and astronomy with the text of the very elementary, and, largely free of mathematics, Tractatus de Sphaera of Johannes de Sacrobosco (c. 1195–c. 1256).
This underwent a radical change in the sixteenth century. The Reformation started by Martin Luther meant that the Lutherans had to create a new education system to replace the Catholic schools and universities, this task was taken up by Philip Melancthon (1497–1560), who became Luther’s Preceptor Germania, Germany’s Schoolmaster.
He set up schools and universities and even introduced a new type of school between the traditional Latin schools and the universities that would become Germany’s modern gymnasia (high schools). Melanchthon had studied under Johannes Stöffler (1452–1531) in Tübingen and as a result had become an ardent believer in astrology.
To do astrology you need astronomy and to do astronomy you need mathematics so, all Melancthon’s schools and universities had a strong integrated mathematics department. Georg Joachim Rheticus (1514–1574) and Erasmus Reinhold (1511–1553), who both played a significant role in the so-called astronomical revolution, were appointed professors for respectively the lower mathematics–arithmetic and geometry–and the higher mathematic–music and astronomy–by Melancthon in Wittenburg.
Addendum 04:04:2024
From an exchange of letters on the topic of astrology
Niels Hemmingsen (1513–1600) to Heinrich Rantzau (1536–1562) (Roskilde, 12 December 1593)
Because the particularly good Philipp Melanchthon, a man of saintly memory (whom we saw and heard teaching in Wittenberg at the same time) understood this, he was an admonisher to his students that they should devote much effort to the study of mathematics so that they would be all the more encouraged to admire God the Creator and Providence.*
It is interesting to note that the first chairs for mathematics at the humanist universities in Northern Italy and Krakow in Poland during the sixteenth century were also basically chairs for astrology because of the rise of astro-medicine or iatromathematics to give it its correct name.
As already noted above, in Catholicism, Christoph Clavius took on the role Melancthon had filled in the Lutheran Church, introducing mathematics as a substantive part of the curriculum in Catholic schools and universities. Clavius’ main motivation was not astrology but astronomy. Clavius trained the teachers, who in turn in a pyramid system trained more teachers. He also wrote excellent up to date textbooks covering all of the main mathematical topic.
It should be clear by now that for various reasons by the end of the sixteenth century the role, status and nature of mathematics in the European universities had undergone a major evolution since the beginning of the century. The foundations had been laid for the mathematics based development of physics that would take place over the seventeenth century. There was another important factor in this development coming from practical mathematics which I will deal with in the next episode in this series.
* Taken from Günther Oestmann, The Viceroy and the Stars: Heinrich Rantzau’s Attitude towards Astrology: A Contribution to Cultural History of the 16th Century to be published