The Clockwork Universe: Isaac Newton, the Royal Society, and the Birth of the Modern World - Edward Dolnick (2011)
Part II. Hope and Monsters
Chapter 33. “Euclid Alone Has Looked on Beauty Bare”
Descartes unveiled his new graphs in 1637, in an appendix to a work called Discourse on Method. The book is a milestone in the history of philosophy, the source of one of the best known of all philosophical maxims. In the Discourse Descartes set out his determination to reject all beliefs that could possibly be incorrect and to build a philosophy founded on indisputable truths. The world and everything in it might be an illusion, Descartes argued, but even if the world was but a dream it was his dream, and so he himself could not be merely an illusion. “I think, therefore I am.”
In the same work he added three short afterwords, each meant to demonstrate the power of his approach to philosophy. In an essay called “Geometry,” Descartes talked about curves and moving points; he explained that a curve can be depicted in a picture or captured in an equation and showed how to translate between the two; he discussed graphs and the use of what are known today as Cartesian coordinates. He understood the value of what he had done. “I do not enjoy speaking in praise of myself,” he wrote in a letter to a friend, but he forced himself. His new, graph-based approach to geometry, he went on, represented a leap “as far beyond the treatment in the ordinary geometry as the rhetoric of Cicero is beyond the ABC of children.”
It did. The wonder is that something so useful and so obvious—in hindsight—should have eluded the world’s greatest thinkers for thousands of years. But this is an age-old story. In the making of the modern world, the same pattern has recurred time and again: some genius conceives an abstract idea that no one before had ever grasped, and in time it finds its way so deeply into our lives that we forget that it had to be invented in the first place.
Abstraction is always the great hurdle. Alfred North Whitehead argued that it was “a notable advance in the history of thought” when someone hit on the insight that two rocks and two days and two sticks all shared the abstract property of “twoness.” For countless generations no one had seen it.
The same holds for nearly every conceptual breakthrough. The idea that “zero” is a number, for instance, proved even more elusive than the notion of “two” or “seven.” Whitehead again: “The point about zero is that we do not need to use it in the operations of daily life. No one goes out to buy zero fish. It is in a way the most civilized of all the [numbers], and its use is only forced on us by the needs of cultivated modes of thought.” With zero in hand, we suddenly have a tool kit that lets us start building the conceptual world. Zero opens the way to place notation—we can distinguish 23 from 203 from 20,003—and to arithmetic and algebra and countless other spinoffs.
Negative numbers once posed similar mysteries. Today the concept of a $5 bill is easy to understand, and so is a $5 IOU. A temperature of 10 degrees is straightforward, and so is 10 degrees below zero. But in the history of the human race, for the greatest intellects over the course of millennia, the notion of negative numbers seemed as baffling as the idea of time travel does to us. (Descartes wrestled to make sense of how something could be “less than nothing.”) Numbers named amounts—1 goat, 5 fingers, 10 pebbles. What could negative 10 pebbles mean?
(Lest we grow too smug we should remember the dismay of today’s students when they meet “imaginary numbers.” The name itself [coined by Descartes, in the same essay in which he explained his new graphs] conveys the unease that surrounded the concept from the start. Small wonder. Students still learn, by rote, that “positive times positive is positive, and negative times negative is positive.” Thus, -2 × -2 = 4, and so is 2 × 2. Then they learn a new definition—an imaginary number is one that, when multiplied by itself, is negative! It took centuries and the labors of some of the greatest minds in mathematics to sort it out.)
The ability to conceive strange, unintuitive concepts like “twoness” and “zero fish” and “negative 10 pebbles” lies at the heart of mathematics. Above all else, mathematics is the art of abstraction. It is one thing to see two apples on the ground next to three apples. It is something else to grasp the universal rule that 2 + 3 = 5.
Reality versus abstraction. Photo of cow, left. Painting of cow by Dutch artist Theo van Doesburg, right, © The Museum of Modern Art/licensed by SCALA/Art Resources, NY.
In the history of science, abstraction was crucial. It was abstraction that made it possible to look past the chaos all around us to the order behind it. The surprise in physics, for instance, was that nearly everything was beside the point. Less detail meant more insight. A rock fell in precisely the same way whether the person who dropped it was a beauty in silk or an urchin in rags. Nor did it matter if the rock was a diamond or a chunk of brick, or if it fell yesterday or a hundred years ago, or in Rome or in London.
The skill that physics demanded was the ability to look past particulars to universals. Just as someone working on a geometry problem would not care whether a triangle was drawn in pencil or ink, so a scientist seeking to describe the world would dismiss countless details as true but irrelevant. Much of a modern physicist’s early training consists in learning to transform colorful questions about such things as elephants tumbling down mountainsides into abstract diagrams showing arrows and angles and masses.
The move from elephants to ten-thousand-pound masses echoes the transformation from Aristotle’s worldview to Galileo’s. The battle between the two approaches was as sweeping as a contest can be, far more than a debate over whether the sun circled the Earth or vice versa, big as that issue was. The broader questions had to do with how to study the physical world. For Aristotle and his followers, the point of science was to engage with the real world in all its complexity. To talk of weights plummeting through vacuums or perfect spheres rolling forever across infinite planes was to mistake idealized diagrams for reality. But the map was not the territory. Explorers needed to grapple with the world as it is, not with a dessicated and lifeless counterpart.
In Galileo’s view, this was exactly backward. The way to understand the world was not to focus on its every quirk and blemish but to look beyond those distractions to the deeper truths they obscured. When Galileo talked about whether heavy objects fall faster than light ones, for instance, he imagined ideal circumstances—objects falling in a vacuum rather than through the air—in order to avoid the complications posed by air resistance. But Aristotle insisted that no such thing as a vacuum could exist in nature (it was impossible, because objects fall faster in a thin medium, like water, than they do in a thick one, like syrup. If there were vacuums, then objects would fall infinitely fast, which is to say they would be in two places at once).42 Even if a vacuum could somehow be contrived, why would anyone think that the behavior of objects in those peculiar conditions bore any relation to ordinary life? To speculate about what might happen in unreal circumstances was an exercise in absurdity, like debating whether ghosts can get sunburns.
Galileo vehemently disagreed. Abstraction was not a distortion but a means of seeing truth unadorned. “Only by imagining an impossible situation can a clear and simple law of fall be formulated,” in the words of the late historian A. Rupert Hall, “and only by possessing that law is it possible to comprehend the complex things that actually happen.”
By way of explaining what the abstract, idealized world of mathematics has to do with the real world, Galileo made an analogy to a shopkeeper measuring and weighing his goods. “Just as the accountant who wants his calculations to deal with sugar, silk, and wool must discount the boxes, bales and other packings, so the mathematical scientist … must deduct the material hindrances” that might entangle him.
The importance of abstraction was a crucial theme, and Galileo came back to it often. At one point he exchanged his shopkeeper image for a more poetic one. With abstraction’s aid, he wrote, “facts which at first sight seem improbable will … drop the cloak which has hidden them and stand forth in naked and simple beauty.”
Galileo won his argument, and science has never turned back. Mathematics remains the language of science because, ever since Galileo, we have taken for granted that abstraction is the pathway to truth.