Before Mathematicians Stopped Being Philosophers
There was a time, not too long ago, when mathematicians called themselves philosophers. Only they were so much more: a quaint band of astronomers, humanists, alchemists, cardinals, book hunters, and of course, artists.
Yet, for early modern natural philosophers as for contemporary mathematicians, the purpose of the inquiry remained the same: to fathom the relations and patterns behind nature’s behaviour. This resolution, rooted in the sixteenth century with the Galilean method (a sequence of steps arranged in order to verify a hypothesis by observational and mathematical means), is still the corner stone of science.
During the Renaissance though, there was considered to be meaning behind each mathemathical pattern. Accordingly, mathematical thought was usually escorted by metaphysical contemplation. It was a relationship cultivated in a generous conversation in which each part kindled the other.
To solve mathematical problems and to contemplate their meaning was thought to be a form of divine proximity. Problems related to geometry and proportion were especially considered expressions of a rapport between the divine and the human mind.
A rational and spiritual endeavour
Nicholas of Cusa was a cardinal, natural philosopher, humanist, theologian and a cracking mathematician. Throughout his work, one finds the use of flexible and rich metaphors with the purpose not of illustrating but of penetrating the subject of his inquiries.
For the former liberal arts student and law doctor, mathematics became an essential and dialogical form of inquiry. We don’t know exactly how and when this recurrent mathematical thinking matured, but it seems to have been mostly self-taught, perhaps initially sparked by his years in Padua, especially by the relationship with his college and lifelong friend Paolo Toscanelli.[1]
Of all the mathematical endeavors led by the cardinal throughout his prolific life, one persisted. For Cusanus, the ancient geometric problem of squaring the circle was not only a practical challenge to be demonstrated, but a source for metaphysical exploration.
The challenge consisted in constructing a square with the same area as a given circle by using only a finite number of steps with compass and straight edge; these were the instruments usually used by geometers and mathematicians for demonstrations since Euclid’s time when the challenge was first proposed. The problem — though finally proven in 1882 to have no solution for the transcendental nature of π — [2] served for centuries as a rich mathematical exercise, a joyful geometrical exploration and a meaningful abstract reflection.
In antiquity the assignment of finding π had been a recurrent challenge all across the Fertile Crescent to India. A tight calculation was given in third century BC by the Syracuse sage Archimedes, who found a solution using the “method of exhaustion” — a premature approximation to infinitesimal calculus — circumscribing it to 310/71 < π < 31/7.[3]
Though eagerly pursued, in early modern Europe the case was still far from being closed. After several attempts, and what was soon proven a false claim, Cusanus shifted from the obstinate search for a solution for π to stating its incommensurability in a transcendental manner. “Transferring the method of exhaustion to theology, Cusanus believed that through awareness of an “inapprehensible precision of the truth,” the human intellect could approach the divine with ever greater propinquity but not gain equality except through a dissolution into it, as an infinitely sided polygon would dissolve into the circle.”[4]
Cusanus contemplated the problem as an expression of the infinite and ineffable nature of the divine in relation to the finite and phenomenological nature of the human mind. In De docta ignoratia, Cusanus used geometry and arithmetic as extended metaphors and evidence of how our intellect and knowledge of created things can only ever be constricted.
The measure of things, the formulation of the way they relate to other magnitudes, even their conceptualization, their nominalization, cannot reveal their true essence; for essence cannot be quantifiable, cannot be measured, cannot be named; it is by its nature ineffable. “For truth is not something more or something less but is something indivisible. Whatever is not truth cannot measure truth precisely. … For the intellect is to truth as an inscribed polygon is to the inscribing circle.”[5]
Plotting the world: the advent of perspective in the imaginary of the Italian Quattrocento
During his several travels and long stays in Rome, between the fiasco of the baptistery doors contest and his final return to Florence to take on the construction of the dome of Santa Maria di Fiore, Filippo Brunelleschi devoted himself to an arcane quest. With a secrecy characteristic of his life and work, Filippo started an archaeological labor. The details of his exact doings during those years in the eternal city remain subject to speculation. All we know is that he and his expedition companion, the artist and lifetime friend Donatello, spent their days digging up, drawing and studying ancient roman architectural and sculptural remains with such concealment and fixation roman citizens were suspicious of their affairs as “treasure hunters”[6] or necromancers.[7]
Measuring heights, weights, depths and circumferences soon led to relating them mathematically in the form of ratios and proportions. And so, the young Filippo slowly started listening to the language of classical architecture, first understanding its vocabulary, then deriving the rules of its syntax, and finally speaking its language in a stunning creative manner.
From the results of his calculations Filippo recognized the account behind classical architectural aesthetics, especially the distinctive character of the architectural orders Doric, Ionic and Corinthian. The key was to spell out how the parts relate to each other and with the whole,[8] an idea from antiquity explicitly found in Euclid’s Elements and stamped in the spirit of Renaissance art and thought.[9]
Euclidean geometry was not strange for a young Florentine. Though the Elements (first published c. 300 BC) was not printed until 1482, hand written copies of it, along with Leonardo Fibonacci’s Practica Geometriae (written in1220), were studied and taught as part of the liberal arts program in the schools of north Italy. Euclid’s concepts became a fundamental part in the dissertations on art, artists started to undertake from the late quattrocento onwards. After all, Euclidean geometry was a visualized mathematical tradition.
The definitions for point, line or plane offered the artist a conceptual nomenclature to think about form and representation in abstract terms, so they could be fairly returned to the phenomenological realm by the art of craftsmanship. The study of Euclidean geometry was the ferment artists needed for the re-emergence of perspective into the imaginary of the world.
In this enterprise, artists turned into philosophers of art; an unbeatable twosome in which creation and contemplation of the creative process happened concurrently.
Whereas those who followed preferred theory to explain the principles of perspective, Brunelleschi was a man of action. As early as ca.1414, he presented the first account of the artifice since antiquity in a public performance in Piazza di San Giovanni on returning from one of his visits the eternal city. How compelling and unforgettable he must have been! For however transitory the nature of a public performance, the message was not forgotten, but rather took root and flourished among his peers.
Written between 1435 and 1436, Leon Battista Alberti dedicated the vernacular edition of Della Pittura, a dissertation on the principles of painting and perspective, to Brunelleschi. In this book, Alberti used Euclidean definitions of point, line and surface to articulate a system for the practice of painting.[10] Alberti begins his treatise by acknowledging the mathematical substrate of his work:
“To make clear my exposition in writing this brief commentary on painting, I will take first from the mathematicians those things with which my subject is concerned. When they are understood, I will enlarge on the art of painting from its first principles in nature in so far as I am able.”[11]
Even when Alberti clarifies that the mathematical principles in his book are only for the purpose of craftsmanship and representation and insisted on being regarded only as a painter and not a mathematician, the study of the history of mathematics should take seriously early modern artists’ commentaries and dissertations.[12]
By bringing depth into a two-dimensional plane and articulating a visual language based on harmonic proportions, the artist of the quattrocento made beauty and numbers speak to each other. Visuality finally formalized its long relationship to comprehension. For now, as never before, to create a picture whether mental or physical, to be able to see, became equivalent to understanding.
Artists’ contemplation of the qualities of beauty and art was intrinsically related to numerical relations. The different parts of a painting or a building were nothing if not related to the whole. Nothing was left alone; to be beautiful and true, a work of art needed to be a world of its own, a harmonic conversation of all the parts.
Nature itself seemed to be emerging out of numerical relations and patterns. Whether found hidden in the form of rivers, trees, leaves, lightning bolts, veins, or in nature’s behavior, mathematics became the vehicle to inquire into the manifested world.
The emergence of the concept of gravity
Two paths were about to diverge from this notion. One was forever confident in the certitude of reasoning, controlled experimentation and empirical demonstration, a position openly set to discredit others outside its own. Another, more flexible, was open to daring explorations from rational reasoning to ancient magical thought.
Perhaps like no other time, early modernity shows how thought promoted by intuition and imagination, hosting and confronting different perspectives and approaches in dialogical coalition, brought critical breakthroughs. This was the case for the emergence of the notion of gravity.
Still today modern science doesn’t fully understand gravity. It can describe its effects, what it does and how it does it, but its nature keeps shifting at every attempt to define it. For Johannes Kepler’s it was an attraction force that kept planets elliptical trajectories around the sun. For Isaac Newton it was a universal law of attraction, described in the equation F=G Mm/r2, present for all mass. For Einstein it was not law but the curvature of space time, the particular geometry of the fabric of the cosmos determine the trajectories. For quantum mechanics, it is still a challenge applying gravity into their equations, since subatomic particles have infinitesimal small mass, so that the force of gravity as we understand it doesn’t interfere with them.
But how all this journey around the concept of gravity began? One can only try to imagine how the first notion of gravity emerged in Johannes Kepler’s spirited mind. He didn’t name it gravity back then. For him it was the bold answer to the question of how planets “know” where to go and how they keep their fixed elliptical trajectories around the sun. The answer was that they were guided by a magnet-like force of attraction related to their mass, the same force responsible on Earth for the change of tides, due to the attraction of the moon.
This idea turned out to be key in the formulation of Laws of Planetary Motion, published by Kepler between 1609 and 1619, and still used today for launching spacecrafts or in the calculations of planetary trajectories with stunning accuracy.
Kepler’s contemporary, the illustrious Florentine, natural philosopher and mathematician, Galileo Galilei, never considered Kepler’s notion of attraction despite their correspondence. The subject appeared to Galileo ungrounded, something more suited to the mind of a sorcerer. Furthermore, it lacked empirical demonstration at that time, and therefore could not be taken seriously as truth. Galileo’s own explanation of the tides being based on Earth’s movement around its axis was shortly proven wrong in favor of Kepler’s explanation involving the moon’s so-called “attraction”.
A few decades later, the notion of attraction grew widely in the restless mind of Isaac Newton. He named it Gravity and formulated its modern meaning as a universal law in Philosophiæ naturalis principia mathematica. Both Kepler and Newton shared like minds and forms of inquiry. They welcomed contrasting forms of knowledge and images of the world, from Hermeticism to Neoplatonism, or Arabic and Greek mathematical traditions. Those fertile coalitions resulted in the birth of notion of Gravity.
One wonders why a mind like Galileo dismissed the ideas shared by the young Kepler in his letters. It was after all perfectly in tune with Galileo’s questions at that time. What findings might have grown out of it? One can only wish.
Afterword
Galileo’s drawings and watercolors of the surface of the moon included in Sidereous Nuncious from observations made through his telescope, are much more than a scientific sketch.
Johannes Kepler, in the absence of the advantage of the telescope, imagined a travel to the moon, and once there, dare to looked back to earth from the perspective of its satellite. The adventure, related in the form of the short story titled Somniun, is considered today an early example of science fiction literature, and perhaps, the first recorded perspective of our planet imagined from outside.
In An Inquiry Concerning Human Understating, the Scottish empiricist David Hume, as late as the eighteenth century, referred to Isaac Newton not as a mathematician, neither a physicist, but as a philosopher: “Till a philosopher, at last, arose, who seems, from the happiest reasoning, to have also determined the laws and forces, by which the revolutions of the planets are governed and directed.”[13] The life and thought of Newton was indeed populated by a huge range of topics from Euclidean Geometry to the Corpus Hermeticum, exegetical studies, alchemy, just to mention a few. A revision of his fascinating notebooks (1661–65) immediately show a restless mind working and considering all the different images of the world. It might be only because of this, and not despite of it, that his mind was able to formulate such colossal breakthroughs.
Mathematical thought also played a fundamental role within the renaissance humanistic project. Early modern art sought the essence of beauty and truth in mathematical relationships. In Alberti’s words, “the very same numbers that cause sounds to have that concinnitas, pleasing to the ears, can also fill the eyes and mind with wondrous delight”.[14]
Since then, ever more specialized knowledge may have produced an exponential growth in human understanding as much as it may have blurred a broader, more imaginative and metaphorical form of thought, responsible not so long ago, for achievements such as the formulation of the Law of Universal Gravity.
Despite the restless efforts of our best minds, the last fifty years have proven unsuccessful at solving the incompatibility of relativity and quantum mechanics. Neither have we found a satisfying interpretation for the puzzling double slit experiment, where the act of observation changes the nature of an electron from particle to wave. In such a situation where consciousness plays such an unavoidable role, science might have to find a way to include awareness as a function of time in the equations.
Only in the collision of multiple images of the world, will breakthroughs occur. Eager, more promiscuous minds flooded with metaphor, imagination and meaning will prove more fertile ground for such ideas to grow. For we know, as the late Mary Midgley wrote, that every time an anxious determination to reach a pole of total meaningless in science arises, such “efforts always end by expressing, not a vacuum of meaning but a different meaning, a different drama and not necessarily a better one.”[15]
[1] An astronomer and mathematician from whom unfortunately, few writings remain. Among them, a letter to the Portuguese Fernando Martíns de Roriz encouraging a western route to India, a letter that some years afterwards found its way to the admiral Christopher Columbus.
[2] A transcendental number is not an algebraic number since it is not the solution of any algebraic equation with rational coefficients. Some of the most known transcendental numbers are π and e.
[3] The method consisted in alternatively circumscribe ever-more-complex regular polygons and then inscribed within a circle; the circle’s circumference can be ever-more-precisely approximated in relation to the difference between that of the slightly larger circumscribing polygon and the slightly smaller inscribed polygon, whose measurements can be calculated through rational means. Archimedes himself finally became exhausted after 96-sided polygons. In Perry Brooks. “Circling the Square: The meaningful use of Φ and Π in the Paintings of Piero della Francesca” In Visual Culture and Mathematics in the Early Modern Period, ed. Ingrid Alexander-Skipnes, New York and London, Routledge, 2007, p 87.
[4] Perry Brooks. “Circling the Square: The meaningful use of Φ and Π in the Paintings of Piero della Francesca” In Visual Culture and Mathematics in the Early Modern Period, ed. Ingrid Alexander-Skipnes, New York and London, Routledge, 2007, p 87.
[5] Nicholas of Cusa, De Docta Ignorantia, I, 3–4, trans. Jasper Hopkins, The Arthur J. Banning Press, Minneapolis, Minnesota 1985, p 8. Translated from De docta ignorantia. Die belehrte Unwis- senheit, Book I, ed. Paul Wilpert, Hamburg, Felix Meiner, 1970.
[6] Manneti, Antonio. Life of Brunelleschi, ed. Howard Saalman, trans. Catherine Enggass, University Park, Pennsylvania State University Press, 1970, pp. 52–54.
[7] Filippo and Donatello visit to Rome for the purpose of studying ancient roman architecture a sculpture sparked the beginning of a new kind of expedition to the eternal city, artists started to undertook as a fundamental part of the learning of the arts. With time the practice grows in popularity an became a fundamental trip in the formation of any gentleman, it was the beginning of The Grand Tour.
[8] From this aesthetical-mathematical survey, came out the proportions and ratios hidden behind the outlook of classical architecture. For instance, the length of a Corinthian column is ten times its diameter while the height of its entablature is a quarter of the height of the columns on which it stands.
[9] Euclid’s Elements Book I, Common Notion 5, The whole is greater than part. in Thomas L. Heath, The Thirteen Books of Euclid’s Elements, translated from the Greek text by J. L. Heiberg, Euclid’s Elementa, Cambridge University Press, 1908, p 232.
[10] For more on Euclidean concepts of point and line in artists thought see Caroline O. Fowler. “The Point and Its Line. An Early Modern History of Movement” In Visual Culture and Mathematics in the Early Modern Period, ed. Ingrid Alexander-Skipnes, New York and London, Routledge, 2007, pp. 113–129.
[11] Leon Battista Alberti, On Painting. trans. John R. Spencer, New Haven and London, Yale University Press, 1966, p 43.
[12] Non-euclidean geometry developed in early XIX century by Carl Friedrich Gauss, Nikolái Lobachevski, among others, stated that in point in the infinite, parallel lines within the same plane converged.
[13] David Hume. An Enquiry Concerning Human Understanding, New York, Oxford University Press, 2007, p 10.
[14] Leon Battista Alberti. De re aedificatoria. On the art of building in ten books. trans. and ed. Joseph Rykwert, Robert Tavernor and Neil Leach, Cambridge, Massachusetts, MIT Press, 1988, p 305.
[15] Mary Midgley. Science and poetry, Routledge Classics, London and New York, 2006, p 44.
