Show notes
Euclid inspired Gothic architecture and taught Renaissance painters how to create depth and perspective. More generally, the success of mathematics went to its head, according to some, and created dogmatic individuals dismissive of other branches of learning. Some thought the uncompromising rigour of Euclid went hand in hand with totalitarianism in political and spiritual domains, while others thought creative mathematics was inherently free and liberal. Transcript Gothic architecture is known for its pointed arches. Unlike round arches like a classical Roman aqueduct for example. Those are semi-circular, but Gothic arches are steeper, pointier. Gothic buildings, cathedrals, have these arches everywhere: windows, doorways, and so on. Gothic arches consist of two circular arcs. You can make it like this. First make a rectangular shape. Like a plain window or door. A boring old rectangle. Now let’s spice it up. Take out your compass, and put it along the top side of the rectangle. Draw two circular arcs going up above the rectangle. Use the top side of the rectangle as the radius, and its two endpoints as the two midpoints of the two arcs you are drawing. The two arcs make a pointed extension of the rectangle. Now you have your Gothic window. If you have your Euclid in fresh memory you will recognize at once that this is precisely the type of construction involved in Proposition 1 of the Elements. Coincidence? No, I don’t think so. The Gothic style of architecture arose in Europe in the early 12th century, within a decade or two of the first Latin translation of Euclid’s Elements. If that’s not cause and effect, it‘s an incredible coincidence. There is little direct documentation about this, but, I am quoting now from Otto von Simson’s book The Gothic Cathedral, “at least one literary document survives that explains the use of geometry in Gothic architecture: the minutes of architectural conferences held in 1391 in Milan. The question debated at Milan is not whether the cathedral is to be built according to a geometrical formula, but merely whether the figure to be used is to be the square or the equilateral triangle. The minutes of one particularly stormy session relate an angry dispute between the French expert, Jean Mignot, and the Italians. Overruled by them on a technical issue, Mignot remarks bitterly that his opponents have set aside the rules of geometry by alleging science to be one thing and art another. Art, however, he concludes, is nothing without science, ars sine scientia nihil est. This argument was considered unassailable even by Mignot’s opponents. They hasten to affirm that they are in complete agreement as regards this theoretical point and have nothing but contempt for an architect who presumes to ignore the dictates of geometry.” So the geometrical ethos was very strong indeed. This hardline view probably softened a bit over time. Renaissance art is more expressive, emotive, more alive, one might say, than this rigid late medieval stuff. That’s if we fast-forward two hundred years from these Gothic conferences about how art is nothing without geometry. Then you have people like Michelangelo who said: “the painter should have compasses in his eyes, not in his hands.” I suppose it means that art should go a little more by feeling and intuition, and not be completely dictated by mathematics. But you still have “compasses in your eyes,” so there’s still a very significant role for geometry, it seems. It is also revealing, perhaps, that Michelangelo thought it was important to point this out at all. I guess there were a lot of artists with compasses in their hands running around back then. Why else would Michelangelo feel the need to criticise that practice? In fact, geometry proved useful to art again, in new ways, in the Renaissance. At this time artists discovered (or perhaps rediscovered) the geometrical principles of perspective. Accurate representation of depth in a painting follows simple geometrical principles. The key construction is that of a tiled floor. Like a chessboard type of pattern of floor tiles, but seen in perspective, so tiles that are further away appear smaller in the picture. There are a lot of tiled floors in Renaissance art, because they are great for conveying a sense of depth. You draw it like this. Draw two horizontal lines: one is where the floor starts, and one is the horizon. Divide the floor line into many equal pieces, representing the size of the tiles. Connect all these points to a fixed point on the horizon. This is because all the parallel rows of the floor will appear to converge in one point, just as whenever you are looking at parallel lines that go off into the distance, such as railroad tracks for instance, they appear to meet at the horizon. Next, draw a second horizontal line representing the other edge of the front row of floor tiles. Now here comes the magic step. Draw the diagonal of the first tile, and extend it. Where this line cuts the other lines you have already drawn, those are all corners of other tiles. This is because of the geometrical principle that a straight line, no matter how you look at it, from whatever angle, will always still be straight. A lot of stuff looks funny in perspective: big things can look small, parallel lines appear to meet, perfectly round things appear to be oval, and so on. Perspective distorts shapes in all kind of ways. But straight lines remain straight lines. That is an invariant in all of this. This is why the diagonal of the first floor tile is also the diagonal of successive tiles. Since that makes a straight line in the real world, it must make a straight line in the picture as well. Once you have this diagonal it is easy to complete the rest of the floor. It looks great. It creates a photorealistic sense of depth. No wonder so many artists chose to set their scenes in locales that just happened to have lots of tiled floors. But the insight is much deeper than this. It’s not really about tiled floors; it’s about the correct perspective representation of depth and size generally. Even if you don’t want a tiled floor in your finished picture, it’s still very useful to draw one in pencil on your canvas as a reference grid. You can use the tiles as a guide to transfer and compare sizes at different depths and distances. Then later you can paint it over with some trees or whatever, so nobody can see the grid anymore; you just used it behind the scenes to get the proportions right. The Greeks probably knew about this stuff, but basically no paintings from antiquity have been preserved. But we know they were skilled artists. One guy is said to have painted grapes so realistically that birds came and pecked at it. It seems the Greeks knew the geometrical principles of perspective and used it to creates scenes for the theatre for example. As Vitruvius says, “by this deception a faithful representation of the appearance of buildings might be given in painted scenery, so that, though all is drawn on a vertical flat facade, some parts may seem to be withdrawing into the background, and others to be standing out in front.” This wasn’t just a party trick to the Greeks. It also had philosophical implications. To Plato they raised profound epistemological conundrums. He was concerned that optical illusion painting has “powers that are little short of magical,” “because they exploit this weakness in our nature,” bypassing “the rational part of the soul.” The solution to this problem, as Plato saw it, was a solid mathematical education. Since “sense perception seems to produce no sound result” with these illusory paintings, “it makes all the difference whether someone is a geometer or not.” “The power of appearance often makes us wander all over the place in confusion, often changing our minds about the same thing and regretting our actions and choices with respect to things large and small.” “The art of measurement,” by contrast, “would make the appearances lose their power” and “give us peace of mind firmly rooted in the truth.” Those are all Plato’s words. A rousing case for mathematics! But Plato perhaps drew his conclusions a step too far, rejecting categorically the role of observational data in science: “there’s no knowledge of sensible things, whether by gaping upward or squinting downward.” Science must be based on “the naturally intelligent part of the soul,” not observation. For example, “let’s study astronomy by means of problems, as we do geometry, and leave the things in the sky alone.” With such attitudes, perhaps it is no wonder that the Greeks excelled more in mathematics than in the sciences. But indeed the threat of optical illusions is a legitimate argument in Plato’s defense. When the principles of perspective were rediscovered in Renaissance Italy, they were again at the heart of scientific developments, but this time on the side of empirical science. Galileo looked at the moon through a telescope and concluded that it had mountains and craters. Of course the image one sees through a telescope is flat. But mountains and craters are revealed by the shadows they cast. This is not necessarily very easy to see, or not necessarily a very evident conclusion. Some scholars have argued that the artistic tradition, and its extensive study of perspective and shadows, was a necessary training for the eye to be able to correctly interpret the telescope data. Is it a coincidence that Galileo the telescopic astronomer came from the same land as the great Italian Renaissance painters? Galileo was born and raised in Tuscany, right where so many of these masters had worked. Maybe only someone immersed in this artistic culture had the right eyes to interpret the heavens. A far-fetched theory, in my opinion, but it’s a nice story. Let’s put aside the art stuff now and look at another theme in how mathematics was received in the early modern world. Namely, the status of mathematics in relation to other fields. Geometry carried a certain authority. This led to many tensions. Let’s jump right into the action, with an eyewitness report from 1703. “There has been much canvassing and intrigue made use of, as if the fate of the Kingdome depended on it.” “On the eve of Newton’s election as president [of the Royal Society], matters had deteriorated to such an extent that various fellows could be restrained only with difficulty from a public exchange of blows (or, in one case, the drawing of swords).” Yikes. So what was this conflict on which “the fate of the Kingdome” depended? It was a battle between the mathematical and the non-mathematical sciences within the Royal Society in London. The “philomats” who identified with Newton thought the non-mathematical sciences were hardly science at all. Botany, geology, stuff like that. They just collect data and write down obvious things. There’s no real thinking involved, no advanced theoretical progress, no genius. Here’s how they put it, when they made the case that Isaac Newton, the great mathematician, ought to be the new president of the society to ensure its intellectual quality: “That Great Man [Newton] was sensible, that something more than knowing the Name, the Shape and obvious Qualities of an Insect, a Pebble, a Plant, or a Shell, was requisite to form a Philosopher, even of the lowest rank, much more to qualifie one to sit at the Head of so great and learned a Body.” So science is divided into two camps: mathematical geniuses like Newton, and then people who just know the names of a bunch of insects. As you can imagine, the other side saw it rather differently. They identified with Francis Bacon, who had complained about “the daintiness and pride of mathematicians, who will needs have this science almost domineer over Physic. For it has come to pass, I know not how, that Mathematics and Logic, which ought to be but the handmaids of Physic, nevertheless presume on the strength of the certainty which they possess to exercise dominion over it.” So mathematicians have an inflated ego. They are so full of themselves that they think they have the right to tell others how to think. Here’s how this point was put in 1700: “The World is become most immoderately fond of Mathematical Arguments, looking upon every thing as trivial, that bears no relation to the Compasse, and establishing the most distant parts of Humane Knowledge, all Speculations, whether Physical, Logical, Ethical, Political, or any other upon the particular results of number and Magnitude. In any other commonwealth but that of Learning such attempts towards an absolute monarchy would quickly meet with opposition. It may be a kind of treason, perhaps, to intimate thus much; but who can any longer forbear, when he sees the most noble, and most usefull portions of Philosophy lie fallow and deserted for opportunities of learning how to prove the Whole bigger than the Part.” So mathematics corrupts mind and soul by fostering delusions of grandeur, and by focusing on obscure technical questions instead of on what is really important. Roger Ascham made a similar point in 1570: “Some wits, moderate enough by nature, be many times marred by over much study and use of some sciences, namely arithmetic and geometry. These sciences sharpen men’s wits over much. Mark all mathematical heads, which be wholly and only bent to those sciences, how solitary they be themselves, how unapt to serve in the world.” Meanwhile, the mathematicians, for their part, thought that an exclusive focus on the merely practical is anti-intellectual and beneath a true thinker. Others scientists may use basic mathematics, but the real accomplishment is to understand it. Mathematician William Oughtred put it like this: “The true way of Art is not by Instruments, but by demonstration. It is a preposterous course of vulgar Teachers, to beginne with Instruments, and not with the Sciences, and so to make their Schollers onely doers of tricks, and as it were jugglers.” Very relatable for a modern mathematics teacher. Students are so dependent on calculators that they are “onely doers of tricks.” That’s what you get when mathematics is not respected as an end in itself, but only as a tool for what is practically useful. There’s an interesting twist to this story though. Part of what these opponents of mathematics were attacking was the pedantic focus on theoretical subtleties. Instead of tackling real problems, mathematicians sit around and muse about nuances of definition and postulates that only matter for very subtle foundational debates, not for actual problem solving. A valid critique, you might say, after reading Euclid with all his foundational pedantry. But here’s the twist: Many mathematicians didn’t like that stuff either. Many mathematicians in the 17th century felt that the Greek geometrical style was much too formal. They recognized the value of the Euclidean style for foundational investigations, but they felt that creative mathematics must be much more free and loose. Here’s how Clairaut put it in the 18th century: “[Euclid’s] geometry had to convince stubborn sophists who prided themselves on refusing [to believe] the most evident truths. It was necessary then that geometry have the help of forms of reasoning to shut the idiots up. But times have changed. All reasoning which applied to that which good sense knows in advance is a pure loss and serves only to obscure truth and disgust the reader.” This fits pretty well with what we have said about the Greek context. Euclid’s special style of geometry arose in a critical philosophical climate. Mathematicians had to anticipate attacks from philosophers who wanted to undermine the entire notion that geometrical reasoning was a rigorous way of finding truth. Without this external pressure from philosophy, mathematicians may have been happy with a much more informal style, as they were in other cultures and societies. And as indeed they became again in the 17th century. Almost all mathe…
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