 Hi, I'm Peter Adamson and you're listening to the History of Philosophy podcast, brought to you with the support of the Philosophy Department at King's College London and the LMU in Munich, online at historyoffilosophy.net. Today's episode, The Measure of All Things, Renaissance Mathematics and Art. My grandfather on my father's side was a brilliant engineer who designed jet engines. His brother built his own plane by hand in his garage and his sister had a PhD in biochemistry. My grandmother and her sisters all had degrees in mathematics, and then there's my father who has always loved numbers just as much as he's hated vegetables. He worked in computing, having been a math prodigy who won statewide competitions as a high school student. Once my twin brother and I received phone calls from him on the same day to congratulate us on being exactly 33 and a third years old. I have an aunt who's a wizard at business administration and my non-existent sister is an expert on imaginary numbers, so it would be fair to say that mathematics runs in my family, but it ran right around me. My feeling about math is much like my feeling about using a motorcycle to jump over a row of burning cars. Amazing, wondrous even, but something I'd just as soon leave to other people. Rather than reflect upon my failure to carry on a family tradition, I comfort myself by telling myself that I'm in good company. Many philosophers have admired mathematics while failing to work seriously at it themselves. Aristotle, for example, wrote no technical treatises on geometry, astronomy, or music, but his posterior analytics, which we just saw taking center stage in the methodological theories of Zabarella, is full of examples involving triangles. And the reason is not far to seek. Mathematics seems to offer the ultimate example of certain rigorous human knowledge. If you ask someone to name something that they are most definitely sure about, they're likely to give an example like 2 plus 2 equals 4. And back in the Renaissance, people felt the same way. The 16th century mathematician Giovanni Battista Benedetti wrote a treatise called On Mathematical Philosophy, which called on Aristotle's authority in proclaiming the absolute certainty of this discipline. And if mathematics is truly philosophical, as suggested by Benedetti's title, then any philosopher worth their salt would have to get far beyond the level of 2 plus 2 equals 4. Nowadays, people tend to think of the humanities as roughly the academic disciplines that don't involve numbers. But the original humanists thought of mathematics as a central part of ancient philosophical wisdom. Or at least some of them did. Leonardo Bruni was not one of them. Sounding not unlike me at the age of 15, he gave the excuse that the subtleties of arithmetic and geometry are not worthy to absorb a cultivated mind. But for the most part, humanists were eager to study manuscripts of Archimedes, Euclid, and other ancient mathematicians. Such works took pride of place in Renaissance libraries. Lorenzo de' Medici, for instance, collected manuscripts of Euclid and Theon of Alexandria, and for the work on mechanics, ascribed falsely to Aristotle. Italy was an epicenter for mathematical knowledge in the 15th century, as we can see from the fact that intellectuals from elsewhere in Europe came there to study and get access to texts that were unavailable elsewhere. Take the astronomer Reggio Montanus, who came to Italy from Vienna and met a who's who of humanist scholars, Bessarion, Alberti, Theodor Gaza, Nicholas of Cusa, and George Trapezuntius. These visitors even got into the spirit of humanism by joining in the petty feuds that so enlivened the era. As a devotee of Bessarion, Reggio Montanus dutifully attacked the translation and commentary that Bessarion's rival Trapezuntius devoted to this central work of ancient astronomy, Ptolemy's Almagest. Equally in the spirit of the age was the rhetoric of recovery and revival that surrounded the philological study of ancient mathematics. In the 16th century, by which time key works of mathematics were available in printed editions, scholars were still boasting that they had rescued this discipline from its formerly perilous state. Raphael Bombelik proclaimed, I have restored the effectiveness of arithmetic, imitating the ancient writers. As usual, such self-congratulation went together with denigration of the achievements of the medieval era. The scholastic calculators who applied mathematical concepts to physics in the 14th century were, as I've mentioned before, studied in the Italian universities, but the humanists were for the most part not impressed. In this case, Bruni was more representative when he said that names like Hatesbury, Ockham, and Swains had filled him with horror. As much as they could, the humanists sought to trace mathematical insight and innovation to the ancient Greeks. But they had to admit that progress had been made in the medieval period, especially in the Islamic world. There, Al-Khwarizmi made breakthroughs in algebra, Ibn al-Haytham, Latinized as Al-Hazen, gave the most accurate account of optics to date, and astronomy was brought to new heights of sophistication. This was recognized in such works as Lives of the Mathematicians, written by Bernardino Baldi in imitation of Diogenes Laertius's Ancient Lives of the Philosophers. Baldi, whose name calls to mind another characteristic that runs in my family, was one of several interconnected mathematicians in the 16th century active in the city of Urbino. The founding figure was Federico Comandino, whom Baldi predictably enough credited with having returned ancient mathematics to light, dignity, and splendor. A student of Comandino, Guido Baldo dal Monte, could not help agreeing, saying that his master had written commentaries on Archimedes that smell of the mathematician's own lamp. Comandino wrote on pure mathematics, as well as applied topics like sundials and calculating a body's center of gravity. His successors followed suit. Guido Baldo anticipated Galileo's famous analysis of projectile motion as having the form of a parabola. He even proposed a nice experiment for establishing this. If you cover a ball with ink and roll it up a blank, inclined surface, you'll see that the track it makes is shaped like an arch. Guido Baldo and Baldi were also devoted to the study of mechanics. They thought that Archimedes had worked out the mathematical details of theories that could be found in more schematic form in the supposedly Aristotelian mechanics. As Baldi put the point, Archimedes followed completely in the footsteps of Aristotle as far as the principles were concerned, adding however the refinement of the proofs. The study of mechanics showed how powerful it could be to combine mathematics with empirical observation, precisely the combination that would very shortly be enabling Galileo to make his profound scientific discoveries. It was also useful in practical terms, as we can see from the example of clock building, which transformed perceptions of time during the Renaissance. Imagine experiencing the transition from keeping time by the motions of the sun to having bells mark the time from church towers in your city. Excellent! It was now possible to be late to meetings. If you lived in Bologna, you'd have the humanist Bessarion to thank for this, since he collaborated on the construction of an astronomical clock there. The result was that, as never before, time was money. Already in 1353, Petrak had spoken of the price of time, and in his writings on household economics, Leon Battista Alberti encouraged his readers to be thrifty with their days and hours. Time is a most precious thing, and needs to be spent as efficiently as possible. I avoid sleep and idleness about Alberti, and I am always doing something. The name of Alberti takes us inevitably to another, more famous application of mathematics, the visual arts. Before you listen to the last several dozen episodes of this podcast, this would probably have been the first thing to come to your mind upon hearing the phrase Italian Renaissance. Even if you've never set foot in a museum, you'll have seen images of the sculptures, paintings, and buildings of men like Piero della Francesca, Michelangelo, da Vinci, Brunelleschi, Botticelli, Raphael, and so on, whether as dorm room posters or refrigerator magnets. And there's quite a lot of mathematics in the background of those images, literally. Take da Vinci's Last Supper. You probably know what it looks like, more or less, but you may have to call it up on the internet to notice that the details on the walls and ceilings surrounding the apostles and Christ at the table provide a lovely example of single point perspective. Notice that the lines of perspective converge on Christ's head, a use of geometry in art to make a theological point. Perhaps less familiar is the painting called Tribute Money by Masaccio. It shows Christ surrounded by a circle of figures, again literally, in that he is the center of that circle. The arrangement of figures has both spiritual and aesthetic weight, with the 13 apostles clustered tightly around the savior in a beautifully orchestrated portrayal of physical space. To learn how the effect was achieved, read Alberti. When painting a crowd, he advises, you should put the heads of the figures along the same horizontal line in the painting, but their feet at different lines. This gives the impression that they are the same height, yet standing at different distances from the viewer. It's only one of the many handy tips you can find in Alberti's writings on art, the most important of which are on painting and on the art of building. It's pretty obvious that architecture involves a lot of mathematics, which is one of several reasons I spend my time constructing arguments and not buildings, but perhaps less so with painting. Yet Alberti promises in the preface to his treatise on painting that the first of its three books will be devoted entirely to mathematics, and so it is. He says that the artist should be expert in all the liberal arts, but most especially in geometry, because without an understanding of this discipline, it is impossible to depict space convincingly. In particular, one needs to understand the geometry used in the discipline of optics. Alberti looks back to Euclid by way of Ptolemy and Imnal Haytham, among others, as he explains that eyesight can be modeled as a pyramid, whose apex is at the eye and whose base is at the visible object. The pyramid is considered to be made up of lines, which stand either for visual rays extending from the eye to the object, or for rays bouncing off the object and reaching the eye. For the purposes of art, says Alberti, there's no need to decide between these two theories. The extreme rays, which are the outer bounds of the pyramid, allow eyesight to grasp the outline or shape of the object that is seen. So this is why Alberti is speaking of a pyramid rather than a visual cone, as was often done in treatises on optics. If you're looking at a slice of pizza, the rays are arranged in a pyramid whose base is triangular, not circular. The reason things look smaller when they're further away is that the visual pyramid for a more distant object has a smaller base. Meanwhile, the rays inside the pyramid take on the color of what is seen, like a chameleon. All of this is just Alberti's account of normal vision. In the case of a painting, we have to imagine the surface of the picture as a meeting between the visual pyramid, whose apex is at the eye, and a pyramid of rays coming from the virtual world of the painting, whose apex is the vanishing point of perspective. Without getting into further details, you can see how some fairly serious geometry is going to be involved in getting the painting right. Getting the correct representation involves working out what mathematicians call a section, like conic sections in the case of a cone, or what Alberti calls a certain cut of the pyramid. In the case of a pavement or a wall with square panels, like the one in da Vinci's Last Supper, you can actually do the geometry with a straight edge. But for more complex forms, Alberti gives another useful tip, which is to suspend a diaphanous veil between yourself and the scene to be painted, and mark on the veil where the objects appear on this vertical plane. This can then be used as a pattern for the painting itself. Through such devices, the artist quite literally takes the measure of the subject found in nature. In fact, there's a sense in which the subject of every painting is proportion. This art renders the world in miniature, portrayed on a surface as it seems relative to the human viewer. This, speculates Alberti, may be what the ancient sophist Protagoras meant when he said that man is the measure of all things, that everything we see is measured against our own stature and from our own point of view. Another nice way that Alberti makes the same point is to say that if everything in the universe, including us, were suddenly halved and the image was not the same, everything would still look the same. We see here yet again, the Renaissance fascination with the individual, contrasting the limited perspective of each individual and what we might call the God's eye view, which is from no particular vantage point and would see each thing as it truly is. Along the same lines, if you'll pardon the expression, what we see in the painted image is not pure abstract mathematics, but the use of an abstraction in a particular viewing situation. Alberti understood this. In another treatise on painting, he remarked that the points considered by the artist are a sort of mean between a mathematical point and a quantity capable of measurement, perhaps like atoms. It's been observed that the geometrically ordered space of a perspective painting is a staged imitation of what we might if we were placed squarely before forms all lined up in parallel fashion. Alberti was sufficiently conscious of this artificiality that he went to the trouble of inventing a viewing box that kept the observer at exactly the right distance from the painted image. In the case of architecture too, he realized that the task was to negotiate between the abstract and the concrete. As Anthony Grafton has written in his intellectual biography of Alberti, On the Art of Building seeks above all to strike a balance between universal mathematical proportion and local site-specific adaptation. Alberti's ideal architect, says Grafton, is a godlike figure who imposes a mathematical order on unruly matter. This attitude was one that Alberti learned from his favorite source, classical antiquity. In particular, he took inspiration from the architectural work of Vitruvius and even divided his own treatise into 10 books in imitation of Vitruvius. But there was another more obviously philosophical ancient influence at work, namely Platonism. Alberti frequented Ficino's circle of Platonists in the 1460s and was called a Platonic mathematician by Ficino himself. Coming from him, that was obviously a great compliment. Platonism gave architects a way to think about their application of abstract forms to concrete buildings, as when they designed churches as a half-sphere, that is a dome, over a cube-shaped interior. This could be taken to represent heaven vaulting above the earth, but it was also a way of giving two of the five geometrical solids, mentioned in Plato's Timaeus, a more literal kind of solidity. And, even as Platonism was inspiring the architects, architecture was inspiring the Platonists. In his dialogue on love, Ficino explains the doctrine of Platonic forms by comparing it to the way the plan of a building appears in the mind of the architect before it is realized in stone. To grasp the idea itself, you must simply imagine that you subtract the matter mentally, but leave the design. This sentiment echoes what we find in Alberti's treatise on architecture, when he writes about drafting the plan for a building as a precise and correct outline conceived in the mind, made up of lines and angles, and perfected in the learned intellect and imagination. And, just to confirm the parallel between applied mathematics in the visual arts and in mechanics, it's worth quoting the aforementioned Bernardino Baldi, who wrote that not all mathematical proofs apply to quantities separated from matter. Sometimes, such proofs are adapted to sensible objects and demonstrate the marvelous effects which occur in them. Of such sort are the proofs in perspective and mechanics. When Renaissance men like Alberti, Ficino, and Baldi traced such ideas back to the classical world, they found that the trail did not end with Vitruvius or Archimedes or even Plato. It ended with Pythagoras. This shadowy, indeed nearly mythical, presocratic philosopher was often held up as a moral exemplar, and was also the ultimate authority for the idea that the cosmos is fundamentally mathematical. Pythagoreanism ran deep in Renaissance humanism and Platonism. It manifested in everything from the circular design of those utopian cities we discussed a few episodes back, to Ficino's excitement over the fact that Plato died on his own birthday and at the age of 81, which is 9 squared, to Pico della Mirandola's choice to defend exactly 900 theses at Rome, the number, he said, of the excited soul. The mathematician Baldi went so far as to compose a lengthy biography of Pythagoras, whom he called the prince of Italian philosophy and inferior to God, but superior to all other men. To think like a Pythagorean meant discerning mathematical structures everywhere in nature and even beyond nature. For a Pythagorean portrayal of the natural world, you cannot beat On the Harmony of the Cosmos, written in 1525 by Francesco Giorgio or Zorzi. This work is influenced by Ficino's understanding of the history of philosophy and looks back to themes of universal harmony found in both ancient Platonists and biblical sources. For a Pythagorean portrayal of the supernatural world, meanwhile, there's Luca Pacioli's 1509 work On Divine Proportion, published with illustrations by none other than Leonardo da Vinci. Pacioli was both an accomplished mathematician and a religious preacher, and wished to show that the divine trinity can be understood in geometrical terms. Take for instance the golden section, a line divided so that the ratio of its shorter segment to its longer segment is the same as the ratio of the longer segment to the whole line. Pacioli suggests that the two segments and the whole line are a fitting image of the trinity, especially since the ratio at work is an irrational number and thus undefinable, like God himself. Pythagoras's influence also made itself felt, or rather seen, in the visual arts. Take the urban fantasy scape ascribed to Fra Can Nabbale, called the Ideal City. It's the ultimate distillation of the Renaissance fascination for classicism and mathematics into a single image. Or, check out what may be the most familiar visual representation of philosophy ever created, Raphael's School of Athens. In the middle, famously, are Plato and Aristotle, Plato pointing to the heavens and Aristotle with his hand held flat, symbolizing that virtue is a mean. But ignore them for now, and notice instead two figures towards the front of the scene, dominating the left and the right groups. They are Pythagoras and Euclid, the former writing in a book and representing arithmetic, the latter poised above a tablet with a compass and representing geometry. I think they really hold the whole thing together. Despite all this enthusiasm for mathematics and despite the excitement that Pythagoras produced in Philosophical Souls, the discipline did not have the most secure of footings in Renaissance institutions. Comandino complained that mathematics was insufficiently present at the universities, and with some justice. There were some positions for mathematical instruction, but these were not so numerous as those devoted to natural philosophy and medicine. Comandino himself trained originally in medicine, but switched fields when he saw how doctors failed to save the lives of ill family members. Some intellectuals managed to maintain an interest in both fields or even forge links between medicine and the mathematical arts. Alberti emphasized that a well-designed building promotes health, for instance by giving ample and pleasant spaces for taking walks as exercise. And Ficino called Pythagoras an expert in both medicine and music, in part on the grounds that music can be used therapeutically to maintain and restore health. In a couple of episodes, we'll be turning to the figure who, more than any other Italian Renaissance thinker, made a name for himself in both of these fields, Girolamo Cardano. But first, I'll be taking a more general look at medicine and philosophy in this period. I hope I can count on you joining me for that, here on The History of Philosophy, without any gaps.