What We Cannot Know. Marcus du Sautoy

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СКАЧАТЬ doesn’t matter how many times in a row you get a 6: this has no influence on what the dice is going to do on the next throw.

      So is there some way of knowing how my dice is going to land? Or is that knowledge always going to be out of reach? Not according to the revelations of a scientist across the waters in England.

       THE MATHEMATICS OF NATURE

      Isaac Newton is my all-time hero in my fight against the unknowable. The idea that I could possibly know everything about the universe has its origins in Newton’s revolutionary work Philosophiae Naturalis Principia Mathematica. First published in 1687, the book is dedicated to developing a new mathematical language that promised the tools to unlock how the universe behaves. It was a dramatically new model of how to do science. The work ‘spread the light of mathematics on a science which up to then had remained in the darkness of conjectures and hypotheses’, declared the French physicist Alexis Clairaut in 1747.

      It is also an attempt to unify, to create a theory that describes the celestial and the earthly, the big and the small. Kepler had come up with laws that described the motions of the planets, laws he’d developed empirically by looking at data and trying to fit equations to create the past. Galileo had described the trajectory of a ball flying through the air. It was Newton’s genius to understand that these were examples of a single phenomenon: gravity.

      Born on Christmas Day in 1643 in the Lincolnshire town of Woolsthorpe, Newton was always trying to tame the physical world. He made clocks and sundials, constructed miniature mills powered by mice, sketched countless plans for buildings and ships, and drew elaborate illustrations of animals. The family cat apparently disappeared one day, carried away by a hot-air balloon that Newton had made. His school reports, however, did not anticipate a great future, describing him as ‘inattentive and idle’.

      Idleness is not necessarily such a bad trait in a mathematician. It can be a powerful incentive to look for some clever shortcut to solve a problem rather than relying on hard graft. But it’s not generally a quality that teachers appreciate.

      Indeed, Newton was doing so badly at school that his mother decided the whole thing was a waste of time and that he’d be better off learning how to manage the family farm in Woolsthorpe. Unfortunately, Newton was equally hopeless at managing the family estate, so he was sent back to school. Although probably apocryphal, it is said that Newton’s sudden academic transformation coincided with a blow to the head that he received from the school bully. Whether true or not, Newton’s academic transformation saw him suddenly excelling at school, culminating in a move to study at the University of Cambridge.

      When bubonic plague swept through England in 1665, Cambridge University was closed as a precaution. Newton retreated to the house in Woolsthorpe. Isolation is often an important ingredient in coming up with new ideas. Newton hid himself away in his room and thought.

      Truth is the offspring of silence and meditation. I keep the subject constantly before me and wait ’til the first dawnings open slowly, by little and little, into a full and clear light.

      In the isolation of Lincolnshire, Newton created a new language that could capture the problem of a world in flux: the calculus. This mathematical tool would be key to our knowing how the universe would behave ahead of time. It is this language that gives me any hope of gleaning how my casino dice might land.

       MATHEMATICAL SNAPSHOTS

      The calculus tries to make sense of what at first sight looks like a meaningless sum: zero divided by zero. As I let my dice fall from my hand, it is such a sum that I must calculate if I want to try to understand the instantaneous speed of my dice as it falls through the air.

      The speed of the dice is constantly increasing as gravity pulls it to the ground. So how can I calculate what the speed is at any given instance of time? For example, how fast is the dice falling after one second? Speed is distance travelled divided by time elapsed. So I could record the distance it drops in the next second and that would give me an average speed over that period. But I want the precise speed. I could record the distance travelled over a shorter period of time, say half a second or a quarter of a second. The smaller the interval of time, the more accurately I will be calculating the speed. Ultimately, to get the precise speed I want to take an interval of time that is infinitesimally small. But then I am faced with calculating 0 divided by 0.

       Calculus: making sense of zero divided by zero

      Suppose that a car starts from a stationary position. When the stopwatch starts, the driver slams his foot on the accelerator. Suppose that we record that after t seconds the driver has covered t × t metres. How fast is the car going after 10 seconds? We get an approximation of the speed by looking at how far the car has travelled in the period from 10 to 11 seconds. The average speed during this second is (11 × 11 – 10 × 10)/1 = 21 metres per second.

      But if we look at a smaller window of time, say the average speed over 0.5 seconds, we get:

      (10.5 × 10.5 – 10 × 10)/0.5 = 20.5 metres per second.

      Slightly slower, of course, because the car is accelerating, so on average it is going faster in the second half second from 10 seconds to 11 seconds. But now we take an even smaller snapshot. What about halving the window of time again:

      (10.25 × 10.25 – 10 × 10)/0.25 = 20.25 metres per second.

      Hopefully the mathematician in you has spotted the pattern. If I take a window of time which is x seconds, the average speed over this time will be 20 + x metres per second. The speed as I take smaller and smaller windows of time is getting closer and closer to 20 metres per second. So, although to calculate the speed at 10 seconds looks like I have to figure out the calculation ⁰⁄₀, the calculus makes sense of what this should mean.

      Newton’s calculus made sense of this calculation. He understood how to calculate what the speed was tending towards as I make the time interval smaller and smaller. It was a revolutionary new language that managed to capture a changing dynamic world. The geometry of the ancient Greeks was perfect for a static, frozen picture of the world. Newton’s mathematical breakthrough was the language that could describe a moving world. Mathematics had gone from describing a still life to capturing a moving image. It was the scientific equivalent of how the dynamic art of the Baroque burst forth during this period from the static art of the Renaissance.

      Newton looked back at this time as one of the most productive of his life, calling it his annus mirabilis. ‘I was in the prime of my age for invention and minded Mathematicks and Philosophy more than at any time since.’

      Everything around us is in a state of flux, so it was no wonder that this mathematics would be so influential. But for Newton the calculus was a personal tool that helped him reach the scientific conclusions that he documents in the Principia, the great treatise published in 1687 that describes his ideas on gravity and the laws of motion.

      Writing in the third person, he explains that his calculus was key to the scientific discoveries contained inside: ‘By the help of this new Analysis Mr Newton found out most of the propositions in the Principia.’ But no account of the ‘new analysis’ is published. Instead, he privately circulated the ideas among friends, but they were not ideas that he felt any urge to publish for others to appreciate.

      Fortunately this language is now widely available and it is one that I spent years learning as a mathematical СКАЧАТЬ